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z|A7V(cet8q0vnwLSbE+b158y8UFi??O3cl^MlkUinz|_tJ(qv@_sKNspploj-T&2= z1<2@4wb%Vn-&dDLTd*IdAEXn{5szFcVm}5_g2>+Q0*Eq#D}FcnOaBBxAyTeR@vV{f zZr#;hw-J5h%R=)i?jPLXe-PllkEfAi?5+&Q*k|FO{ofp-0Qiro<NX#AJIrVbng6`#{0=|^&2|1avy?pZxDX{6~cUc`N?23;#ti|5cy={SW_9r2jt@=|^CVxeML*?=#gT@Jm%u^IGxM HfA0S;9Fn0M diff --git a/probability_1/probability_1.qmd b/probability_1/probability_1.qmd index cb2e2bef..46e3e3ad 100644 --- a/probability_1/probability_1.qmd +++ b/probability_1/probability_1.qmd @@ -33,9 +33,18 @@ jupyter: * Gain familiarity with the Bernoulli and binomial random variables ::: -In the past few lectures, we've examined the role of complexity in influencing model performance. We've considered model complexity in the context of a tradeoff between two competing factors: model variance and training error. +In the past few lectures, we've examined the role of complexity in influencing model performance. We've considered model complexity in the context of a tradeoff between two competing factors: model variance and training error. More specifically, in the last lecture we discussed how the optimal theta after regularization is the following: -So far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven't yet discussed *why* exactly this tradeoff exists. +$$ \hat{\theta} = \text{arg}\underset{\theta}{\text{min}}\ \textbf{Loss}(\theta, \text{data}) + \lambda\ \textbf{Regularizer}(\theta)$$ + +Here, the **Loss** term ensures that the model **fits the data** and the **Regularizer** term ensures that the model **isn't too complex**. The $\lambda$ term is the regularization **hyperparameter** which is tuned using **cross-validation** to prevent **over-fitting** to our data. We also have two options for the regularizer: + +- $L_1$ (absolute) penalty $\rightarrow$ **Lasso Regression** + - Allows for **sparsity** where we can drop features that are less useful +- $L_2$ (squared) penalty $\rightarrow$ **Ridge Regression** + - Allows for **robustness** where the weight is spread out over more features + +The goal is to find a balance between **bias (underfitting)** and **variance (overfitting)**. So far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between the two, but we haven't yet discussed *why* exactly this tradeoff exists. To better understand the origin of this tradeoff, we will need to dive into **random variables**. The next two course notes on probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. In specific, we will cover: @@ -49,7 +58,7 @@ We'll go over just enough probability to help you understand its implications fo Recall the following concepts from Data 8: 1. Sample mean: The mean of the random sample -2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean +2. **Central Limit Theorem**: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean a. is roughly normal @@ -86,7 +95,7 @@ $$X = \begin{cases} X(H) = 1 \\ X(T) = 0 \end{cases}$$ Suppose we draw a random sample $s$ of size 3 from all students enrolled in Data 100. -We can define $Y$ as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is $\{0, 1, 2, 3\}$. +We can define the random variable $Y$ as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is $\{0, 1, 2, 3\}$.

rv @@ -94,7 +103,7 @@ We can define $Y$ as the number of data science students in our sample. Its doma Note that we can use random variables in mathematical expressions to create new random variables. -For example, let's say we sample 3 students at random from lecture and look at their midterm scores. Let $X_1$, $X_2$, and $X_3$ represent each student's midterm grade. +For example, let's say we **sample 3 students at random** from lecture and look at their midterm scores. Let $X_1$, $X_2$, and $X_3$ represent each student's midterm grade. We can use these random variables to create a new random variable, $Y$, which represents the average of the 3 scores: $Y = (X_1 + X_2 + X_3)/3$. @@ -109,9 +118,11 @@ But, what exactly is a distribution? Let's dive into this! To define any random variable $X$, we need to be able to specify 2 things: 1. **Possible values**: the set of values the random variable can take on. -2. **Probabilities**: the set of probabilities describing how the total probability of 100% is split over the possible values. +2. **Probabilities**: the chance that the random variable will take each possible value. + - Each probability should be a real-number between 0 and 1 (inclusive) + - The total probability of all possible values should be 1. -If $X$ is discrete (has a finite number of possible values), the probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$, and probabilities must sum to 1: $\sum_{\text{all } x} P(X=x) = 1$, +If $X$ is discrete (has a finite number of possible values), the probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$, and probabilities must sum to 1: $\underset{\text{all } x}{\sum} P(X=x) = 1$, We can often display this using a **probability distribution table**. In the coin toss example, the probability distribution table of $X$ is given by. @@ -122,8 +133,8 @@ We can often display this using a **probability distribution table**. In the coi The **distribution** of a random variable $X$ describes how the total probability of 100% is split across all the possible values of $X$, and it fully defines a random variable. If you know the distribution of a random variable you can: -- compute properties of the random variables and derived variables -- simulate the random variables by randomly picking values of $X$ according to its distribution using `np.random.choice`, `df.sample`, or `scipy.stats..rvs(...)` +- **compute properties** of the random variables and derived variables +- **simulate** the random variables by randomly picking values of $X$ according to its distribution using `np.random.choice`, `df.sample`, or `scipy.stats..rvs(...)` The distribution of a discrete random variable can also be represented using a histogram. If a variable is **continuous**, meaning it can take on infinitely many values, we can illustrate its distribution using a density curve. @@ -131,7 +142,7 @@ The distribution of a discrete random variable can also be represented using a h discrete_continuous

-We often don’t know the (true) distribution and instead compute an empirical distribution. If you flip a coin 3 times and get {H, H, T}, you may ask —— what is the probability that the coin will land heads? We can come up with an **empirical estimate** of $\frac{2}{3}$, though the true probability might be $\frac{1}{2}$. +We often don’t know the (true) distribution and instead compute an empirical distribution. If you flip a coin 3 times and get {H, H, T}, you may ask — what is the probability that the coin will land heads? We can come up with an **empirical estimate** of $\frac{2}{3}$, though the true probability might be $\frac{1}{2}$. Probabilities are areas. For discrete random variables, the *area of the red bars* represents the probability that a discrete random variable $X$ falls within those values. For continuous random variables, the *area under the curve* represents the probability that a discrete random variable $Y$ falls within those values. @@ -139,27 +150,41 @@ Probabilities are areas. For discrete random variables, the *area of the red bar discrete_continuous

-If we sum up the total area of the bars/under the density curve, we should get 100%, or 1. +If we sum up the **total area** of the bars/under the density curve, we should get 100%, or 1. Continuous random variables are more complex (requiring a probability density function), so take a probability class like Data 140 to learn more. + +One common probability distribution is the **Bernoulli** distribution which is a **binary variable** that takes on **two values**: 0 or 1. For example, + +$$X = \begin{cases} + 1, \text{if the coin lands heads} \\ + 0, \text{if the coin lands tails} + \end{cases}$$ -We can show the distribution of $Y$ in the following tables. The table on the left lists all possible samples of $s$ and the number of times they can appear ($Y(s)$). We can use this to calculate the values for the table on the right, a **probability distribution table**. +The Bernoulli distribution is **parameterized by p** the probability P(X=1), and the outcomes can be seen in the **probability table** below. Note that we use the notation **X ~ Bernoulli(p)** to denote that the **random variable X** has a Bernoulli distribution with parameter p. + +| Outcome | Probability | +| ------- | ----------- | +| 1 | p | +| 0 | 1-p | + + -Rather than fully write out a probability distribution or show a histogram, there are some common distributions that come up frequently when doing data science. These distributions are specified by some **parameters**, which are constants that specify the shape of the distribution. In terms of notation, the '~' means "has the probability distribution of". +Rather than fully write out a probability distribution or show a histogram, there are some common distributions that come up frequently when doing data science. These distributions are specified by some **parameters** (the numbers inside the parentheses), which are constants that specify the shape of the distribution. In terms of notation, the '~' again means "has the probability distribution of". These common distributions are listed below: -1. Bernoulli($p$): If $X$ ~ Bernoulli($p$), then $X$ takes on a value 1 with probability $p$, and 0 with probability $1 - p$. Bernoulli random variables are also termed the "indicator" random variables. -2. Binomial($n$, $p$): If $X$ ~ Binomial($n$, $p$), then $X$ counts the number of 1s in $n$ independent Bernoulli($p$) trials. -3. Categorical($p_1, ..., p_k$) of values: The probability of each value is 1 / (number of possible values). -4. Uniform on the unit interval (0, 1): The density is flat at 1 on (0, 1) and 0 elsewhere. We won't get into what density means as much here, but intuitively, this is saying that there's an equally likely chance of getting any value on the interval (0, 1). -5. Normal($\mu$, $\sigma^2$): The probability density is specified by $\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}}$. This bell-shaped distribution comes up fairly often in data, in part due to the Central Limit Theorem you saw back in Data 8. +1. **Bernoulli**($p$): If $X$ ~ Bernoulli($p$), then $X$ takes on a value 1 with probability $p$, and 0 with probability $1 - p$. Bernoulli random variables are also termed the **"indicator" random variables**. +2. **Binomial**($n$, $p$): If $X$ ~ Binomial($n$, $p$), then $X$ counts the number of 1s in $n$ independent Bernoulli($p$) trials. +3. **Categorical**($p_1, ..., p_k$) of values: The probability of each value is 1 / (number of possible values). +4. **Uniform** on the unit interval (0, 1): The density is flat at 1 on (0, 1) and 0 elsewhere. We won't get into what density means as much here, but intuitively, this is saying that there's an equally likely chance of getting any value on the interval (0, 1). +5. **Normal**($\mu$, $\sigma^2$): The probability density is specified by $\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$. This bell-shaped distribution comes up fairly often in data, in part due to the Central Limit Theorem you saw back in Data 8. ## Expectation and Variance -There are several ways to describe a random variable. The methods shown above —— a table of all samples $s, X(s)$, distribution table $P(X=x)$, and histograms —— are all definitions that *fully describe* a random variable. Often, it is easier to describe a random variable using some *numerical summary* rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a "summary" of how the variable tends to behave, they are *not* random. Instead, think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable. +There are several ways to describe a random variable. The methods shown above — a table of all samples $s, X(s)$, distribution table $P(X=x)$, and histograms — are all definitions that *fully describe* a random variable. Often, it is easier to describe a random variable using some *numerical summary* rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a "summary" of how the variable tends to behave, they are *not* random. Instead, think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable. ### Expectation The **expectation** of a random variable $X$ is the **weighted average** of the values of $X$, where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: @@ -169,7 +194,7 @@ The **expectation** of a random variable $X$ is the **weighted average** of the The latter is more commonly used as we are usually just given the distribution, not all possible samples. -We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram, meaning if we simulate the variable many times, it is the long-run average of the simulated values. +We want to emphasize that the **expectation is a *number*, not a random variable**. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the **center of gravity** of the **probability distribution histogram**, meaning if we simulate the variable many times, it is the **long-run average of the simulated values**. #### Example 1: Coin Toss Going back to our coin toss example, we define a random variable $X$ as: @@ -208,13 +233,13 @@ $$\begin{align} Again, note that $\mathbb{E}[X] = 5.9$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**. ### Variance -The **variance** of a random variable is a measure of its chance error. It is defined as the expected squared deviation from the expectation of $X$. Put more simply, variance asks: how far does $X$ typically vary from its average value, just by chance? What is the spread of $X$'s distribution? +The **variance** of a random variable is a measure of its chance error. It is defined as the **expected squared deviation from the expectation** of $X$. Put more simply, variance asks: how far does $X$ typically vary from its average value, just by chance? What is the spread of $X$'s distribution? $$\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]$$ The units of variance are the square of the units of $X$. To get it back to the right scale, use the standard deviation of $X$: $$\text{SD}(X) = \sqrt{\text{Var}(X)}$$ -Like with expectation, **variance and standard deviation are numbers, not random variables**! Variance helps us describe the variability of a random variable. It is the expected squared error between the random variable and its expected value. As you will see shortly, we can use variance to help us quantify the chance error that arises when using a sample $X$ to estimate the population mean. +Like with expectation, **variance and standard deviation are numbers, not random variables**! Variance helps us describe the **variability** of a random variable. It is the **expected squared error** between the **random variable** and its **expected value**. As you will see shortly, we can use variance to help us quantify the **chance error** that arises when using a sample $X$ to estimate the **population mean**. By [Chebyshev’s inequality](https://www.inferentialthinking.com/chapters/14/2/Variability.html#Chebychev's-Bounds), which you saw in Data 8, no matter what the shape of the distribution of $X$ is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.” @@ -228,8 +253,8 @@ This form is often more convenient to use when computing the variance of a varia ## Proof $$\begin{align} \text{Var}(X) &= \mathbb{E}[(X-\mathbb{E}[X])^2] \\ - &= \mathbb{E}(X^2 - 2X\mathbb{E}(X) + (\mathbb{E}(X))^2) \\ - &= \mathbb{E}(X^2) - 2 \mathbb{E}(X)\mathbb{E}(X) +( \mathbb{E}(X))^2\\ + &= \mathbb{E}[X^2 - 2\mathbb{E}[X]X + \mathbb{E}[X]^2] \\ + &= \mathbb{E}[X^2] - 2 \mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}(X)^2\\ &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \end{align}$$ ::: @@ -278,7 +303,7 @@ We can summarize our discussion so far in the following diagram: ## Sums of Random Variables -Often, we will work with multiple random variables at the same time. A function of a random variable is also a random variable. If you create multiple random variables based on your sample, then functions of those random variables are also random variables. +Often, we will work with multiple random variables at the same time. **A function of a random variable is also a random variable.** If you create multiple random variables based on your sample, then functions of those random variables are also random variables. For example, if $X_1, X_2, ..., X_n$ are random variables, then so are all of these: @@ -291,7 +316,7 @@ For example, if $X_1, X_2, ..., X_n$ are random variables, then so are all of th Many functions of random variables that we are interested in (e.g., counts, means) involve sums of random variables, so let's dive deeper into the properties of sums of random variables. ### Properties of Expectation -Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\mathbb{E}[X] = \sum_{x} x P(X=x)$$ +Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\mathbb{E}[X] = \sum_{\text{all possible}\ x} x P(X=x)$$ From it, we can derive some useful properties: @@ -326,6 +351,20 @@ $$\begin{align} 3. If $g$ is a non-linear function, then in general, $$\mathbb{E}[g(X)] \neq g(\mathbb{E}[X])$$ For example, if $X$ is -1 or 1 with equal probability, then $\mathbb{E}[X] = 0$, but $\mathbb{E}[X^2] = 1 \neq 0$. +Using these properties, we can again better understand how we got from the original definition of variance to the computational definition. + +::: {.callout-tip collapse="false"} +## Proof +$$\begin{align} + \text{Var}(X) &= \mathbb{E}[(X-\textcolor{blue}{\mathbb{E}[X]})^2] \hspace{2em} \text{Constants inside expectations are colored \textcolor{blue}{blue}} \\ + &= \mathbb{E}[X^2 - \textcolor{blue}{2\mathbb{E}[X]}X + \textcolor{blue}{\mathbb{E}[X]}^2] \hspace{2em} \text{Expanding the square}\\ + &= \mathbb{E}[X^2] - \mathbb{E}[\textcolor{blue}{2\mathbb{E}[X]}X] +\mathbb{E}[\textcolor{blue}{\mathbb{E}[X]}^2] \hspace{2em} \text{by Property 2}\\ + &= \mathbb{E}[X^2] - 2 \mathbb{E}[X]\mathbb{E}[X] +\mathbb{E}[X]^2 \hspace{2em} \text{by Property 1}\\ + &= \mathbb{E}[X^2] - 2 \mathbb{E}[X]^2 +\mathbb{E}[X]^2 \hspace{2em} \text{Algebra}\\ + &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 +\end{align}$$ +::: + ### Properties of Variance Let's now get into the properties of variance. Recall the definition of variance: $$\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]$$ @@ -346,9 +385,9 @@ In order to compute $\text{Var}(aX+b)$, consider that a shift by $b$ units does Then, $$\begin{align} \text{Var}(aX+b) &= \text{Var}(aX) \\ - &= E((aX)^2) - (E(aX))^2 \\ - &= E(a^2 X^2) - (aE(X))^2\\ - &= a^2 (E(X^2) - (E(X))^2) \\ + &= \mathbb{E}[(aX)^2] - (\mathbb{E}[aX])^2 \\ + &= \mathbb{E}[a^2 X^2] - (a\mathbb{E}[X])^2\\ + &= a^2 (\mathbb{E}[X^2] - (\mathbb{E}[X])^2) \\ &= a^2 \text{Var}(X) \end{align}$$ ::: @@ -361,7 +400,7 @@ $$\begin{align}

2. Variance of sums of random variables is affected by the (in)dependence of the random variables. -$$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{cov}(X,Y)$$ +$$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$$ $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text{ independent}$$ @@ -391,7 +430,12 @@ We can treat the covariance as a measure of association. Remember the definition $$r(X, Y) = \mathbb{E}\left[\left(\frac{X-\mathbb{E}[X]}{\text{SD}(X)}\right)\left(\frac{Y-\mathbb{E}[Y]}{\text{SD}(Y)}\right)\right] = \frac{\text{Cov}(X, Y)}{\text{SD}(X)\text{SD}(Y)}$$ -It turns out we've been quietly using covariance for some time now! If $X$ and $Y$ are independent, then $\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$. Note, however, that the converse is not always true: $X$ and $Y$ could have $\text{Cov}(X, Y) = r(X, Y) = 0$ but not be independent. +It turns out we've been quietly using covariance for some time now! Correlation (and therefore covariance) measures a linear relationship between $X$ and $Y$. + +- If $X$ and $Y$ are correlated, then knowing $X$ tells you something about $Y$. +- “$X$ and $Y$ are uncorrelated” is the same as “Correlation and covariance equal to 0” +- **Independent $X$ and $Y$ are uncorrelated**, or in other words $\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$, because knowing $X$ tells you nothing about $Y$ +- Note, however, that the converse is not always true: **$X$ and $Y$ could be uncorrelated** (having $\text{Cov}(X, Y) = r(X, Y) = 0$) **but not be independent**. ### Equal vs. Identically Distributed vs. i.i.d @@ -405,45 +449,18 @@ Suppose that we have two random variables $X$ and $Y$: Note that in Data 100, you'll never be expected to prove that random variables are i.i.d. -Now let's walk through an example. Say $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation. +Now let's walk through an example. Say $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation (7).

distribution

-However, $Y = X_1$ has a larger variance. +However, looking at the graphs and running this through simulation, we can see that the left distribution ($Y = 2X_1$) has a larger variance.

distribution

-### Example: Bernoulli Random Variable - -To get some practice with the formulas discussed so far, let's derive the expectation and variance for a Bernoulli($p$) random variable. If $X$ ~ Bernoulli($p$), - -$\mathbb{E}[X] = 1 \cdot p + 0 \cdot (1 - p) = p$ - -To compute the variance, we will use the computational formula. We first find that: -$\mathbb{E}[X^2] = 1^2 \cdot p + 0^2 \cdot (1 - p) = p$ - -From there, let's calculate our variance: -$\text{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = p - p^2 = p(1-p)$ - -### Example: Binomial Random Variable - -Let $Y$ ~ Binomial($n$, $p$). We can think of $Y$ as being the sum of $n$ i.i.d. Bernoulli($p$) random variables. Mathematically, this translates to - -$$Y = \sum_{i=1}^n X_i$$ - -where $X_i$ is the indicator of a success on trial $i$. - -Using linearity of expectation, - -$$\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np$$ - -For the variance, since each $X_i$ is independent of the other, $\text{Cov}(X_i, X_j) = 0$, - -$$\text{Var}(Y) = \sum_{i=1}^n \text{Var}[X_i] = np(1-p)$$ ### Summary * Let $X$ be a random variable with distribution $P(X=x)$. @@ -457,3 +474,4 @@ $$\text{Var}(Y) = \sum_{i=1}^n \text{Var}[X_i] = np(1-p)$$ * $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$ Note that $\text{Cov}(X,Y)$ would equal 0 if $X$ and $Y$ are independent. + diff --git a/probability_2/probability_2.qmd b/probability_2/probability_2.qmd index 5cc77169..27bf2d6d 100644 --- a/probability_2/probability_2.qmd +++ b/probability_2/probability_2.qmd @@ -66,6 +66,34 @@ There are several cases of random variables that appear often and have useful pr * Normal($\mu, \sigma^2$), a.k.a Gaussian * $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{\!2}\,\right)$ +### Example: Bernoulli Random Variable + +To get some practice with the formulas discussed so far, let's derive the expectation and variance for a Bernoulli($p$) random variable. If $X$ ~ Bernoulli($p$), + +$\mathbb{E}[X] = 1 \cdot p + 0 \cdot (1 - p) = p$ + +To compute the variance, we will use the computational formula. We first find that: +$\mathbb{E}[X^2] = 1^2 \cdot p + 0^2 \cdot (1 - p) = p$ + +From there, let's calculate our variance: +$\text{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = p - p^2 = p(1-p)$ + +### Example: Binomial Random Variable + +Let $Y$ ~ Binomial($n$, $p$). We can think of $Y$ as being the sum of $n$ i.i.d. Bernoulli($p$) random variables. Mathematically, this translates to + +$$Y = \sum_{i=1}^n X_i$$ + +where $X_i$ is the indicator of a success on trial $i$. + +Using linearity of expectation, + +$$\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np$$ + +For the variance, since each $X_i$ is independent of the other, $\text{Cov}(X_i, X_j) = 0$, + +$$\text{Var}(Y) = \sum_{i=1}^n \text{Var}[X_i] = np(1-p)$$ + ### Example Suppose you win cash based on the number of heads you get in a series of 20 coin flips. Let $X_i = 1$ if the $i$-th coin is heads, $0$ otherwise. Which payout strategy would you choose? @@ -499,3 +527,4 @@ $$ E\left[(Y(x)-\hat{Y}(x))^2\right] = \sigma^2 + \left(E\left[\hat{Y}(x)\right] - g(x)\right)^2 + E\left[\left(\hat{Y}(x)-E\left[\hat{Y}(x)\right]\right)^2\right] $$ ::: +