Previously… (1/2)

The Binomial PMF and CDF

Let $X \sim Binomial(np)$ where $p$ is the probability of success and $n$ be the number of trials $n$.

PMF:

\[P(X = x; n, p) = \binom{n}{x} p^x (1-p)^{(n-x)} \hspace{5px} \text{ for } x = 0,1,2,\cdots, n\]

Use dbinom in R.

CDF:

\[\begin{align*} F(x) = & P(X \le x; p) \\ = & \sum_{k=0}^x \binom{n}{k} p^k (1-p)^{(n-k)} \\ \end{align*}\]

Use pbinom in R.

Previously… (2/2)

The Geometric PMF and CDF

Let $X \sim Geometric(p)$ where $p$ is the probability of success.

PMF:

\[P(X = x; p) = (1-p)^{x-1} p, \hspace{20px} x = 1, 2, 3, \cdots\]

Use dgeom in R. Note that the input for dgeom is $(x-1)$. So if $x=3$, then you need dgeom(3-1).

CDF:

\[\begin{align*} F(x) = & P(X \le x; p) \\ = & \sum_{k=0}^x (1-p)^k p \\ \end{align*}\]

\[ F(x) = \begin{cases} 1-(1-p)^{x} & \text{ if } x \ge 1 \\ 0 & \text{ if } x < 1 \\ \end{cases} \]

Use pgeom in R.

Example 1 (1/2)

Coin Flipping Game. Suppose you have a coin that comes up H 40% of times and T 60% of times. You get paid $2 if H, and you pay out $1 if T. What do you expect to have after 10 tosses?

The intuition:

For 1 coin flip, you are expected value is \[(2)P(H) + (-1)P(T) = (2)(0.40) + (-1)(0.60) = 0.20.\]
For 10 coin flips, you are expected value is \[(10)(0.20) = 2.\]

You would be expected to gain $2 in 10 tosses on average.

Example 1 (2/2)

In probability notation, we let $X \sim Binom(10,0.40)$ be the discrete random variable of the number of heads in 100 tosses, and $Y \sim Binom(10,0.60)$ be the number of tails. So, the PMFs are given by

\[P_X(X = x; 10, 0.40) = \binom{10}{x} (0.40)^x (0.60)^{(10-x)}\]

\[P_Y(Y = y; 10, 0.60) = \binom{10}{x} (0.60)^x (0.40)^{(10-x)}\]

The expected value is written as

\[\begin{align*} E[2X - Y] = & 2E[X] - E[Y] \\ = & 2 \sum_{x = 0}^{10} x P_X(x) - \sum_{y = 0}^{10} y P_X(y) \\ = & 2(10)(0.40) - (10)(0.60) \\ = & 10(2(0.40)-0.60) \\ = & 2 \\ \end{align*}\]

Expected Values for Discrete Random Variables

[PSDR] Definition 3.8 For a discrete random variable $X$ with pmf $p$, the expected value of $X$ is \[E[X] = \sum_{x} x p(x)\]

provided this sum exists, where the sum is taken over all possible values of the random variable $X$.

[PSDR] Theorem 3.2 (Law of Large Numbers) The mean of n observations of a random variable $X$ converges to the expected value $E[X]$ as $n \to \infty$, assuming $E[X]$ is defined.

Properties of Expected Values

Expected value of a constant:

If $a$ is a constant, then $E[a] = a$.

Scalar multiplication of a random variable:

If $a$ is a scalar constant, then $E[aX] = aE[X]$.

Sums of Random Variables:

If $X_1, X_2, \cdots, X_k$ are $k$ independent random variables, then $E[X_1 + X_2 + \cdots + X_k] = E[X_1] + E[X_2] + \cdots + E[X_k]$.

Expectation of a product of random variables

If $X$ and $Y$ are random variables, then $E[XY] = E[X]E[Y]$.

Expected Values for DRVs