AP Stats combining random variables cheat sheet

Sum of Random Variables

If X and Y are independent, then $\displaystyle E(X+Y) = E(X) + E(Y)$. Example: If X = $\displaystyle 10$ and Y = $\displaystyle 20$, $\displaystyle E(X+Y) = 30$.

True or False: Variance of a sum is sum of variances.

False. For independent variables: $\displaystyle Var(X+Y) = Var(X) + Var(Y)$. If dependent, must consider covariance.

Difference between discrete and continuous variables?

Discrete: Countable outcomes (e.g., number of cars). Continuous: Infinite outcomes (e.g., height in inches).

Fill in the blank: If X ~ N(50, 4), then $\displaystyle P(X < 50)$ is _____?

0.5 - The mean divides the normal distribution.

Expected value of a random variable

Formula: $\displaystyle E(X) = ext{sum of } (x_i * P(x_i))$. Example: For $\displaystyle X = {1,2,3}$, $\displaystyle P = {0.2,0.5,0.3}$, $\displaystyle E(X) = 1*0.2 + 2*0.5 + 3*0.3 = 2.3$.

True or False: The mean of a random variable is always its most likely value.

False. The mean can be skewed by extreme values, while the mode is the most frequent.

Variance of a random variable

Formula: $\displaystyle Var(X) = E(X^2) - (E(X))^2$. Example: If $\displaystyle E(X) = 3$ and $\displaystyle E(X^2) = 12$, then $\displaystyle Var(X) = 12 - 9 = 3$.

Question: What happens when you combine independent normal variables?

The result is also normally distributed. If $\displaystyle X ext{ and } Y$ are $\displaystyle N( ext{mean}, ext{variance})$, then $X + Y ext{ is } N( ext{sum of means}, ext{sum of variances}).

Cause → Effect: Increasing probability of an event _____ the expected value.

Increases - Higher probabilities lead to higher expected outcomes.

Example of a binomial random variable

Example: Flipping a coin 10 times. Let X = number of heads. X follows a Binomial distribution B(n=10, p=0.5).

Difference: Binomial vs. Geometric

Binomial: Fixed number of trials; Geometric: Number of trials until first success. Example: Flipping until the first head.

Calculate E(X) for a fair die

E(X) = $\displaystyle 1/6(1) + 1/6(2) + 1/6(3) + 1/6(4) + 1/6(5) + 1/6(6) = 3.5$.

Sum of independent random variables?

If $\displaystyle X$ and $\displaystyle Y$ are independent, then $\displaystyle E(X + Y) = E(X) + E(Y)$.

True or False: Var(X + Y) = Var(X) + Var(Y)

True, if $\displaystyle X$ and $\displaystyle Y$ are independent.

Variance of a sum: X + Y

If $\displaystyle X$ and $\displaystyle Y$ are independent: $\displaystyle Var(X + Y) = Var(X) + Var(Y)$.

Example of combining means.

A store sells two items: Item A: $\displaystyle E(A) = 20$ Item B: $\displaystyle E(B) = 30$ Total expected sales: $\displaystyle E(A + B) = 50$.

Fill in the blank: E(aX) = ...

E(aX) = aE(X), where $\displaystyle a$ is a constant.

Difference: Binomial vs. Geometric.

Binomial: fixed trials, $\displaystyle k$ successes. Geometric: until first success, 1 success.

Example: Random variable sum.

If $\displaystyle X$ (dollars earned) = 50, $\displaystyle Y$ (dollars saved) = 100, $\displaystyle E(X + Y) = 150$.

True or False: Covariance measures independence.

False; covariance indicates direction of linear relationship, not independence.

What is the mean of 2X + 3?

If $\displaystyle E(X) = 10$, then $\displaystyle E(2X + 3) = 2E(X) + 3 = 23$.

Sum of two variables: Y = X + Z.

If $\displaystyle E(X) = 15$, $\displaystyle E(Z) = 25$, then $\displaystyle E(Y) = 40$.

Standard deviation of scaled variable?

If $\displaystyle Y = aX$, then $\displaystyle SD(Y) = |a|SD(X)$.

Fill in the blank: Var(aX) = ...

Var(aX) = a^2Var(X), where $\displaystyle a$ is a constant.