AP Statistics Chapter 16

Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. Value of x x1x1 x2x2 x3x3 ……xkxk Probabilit y p1p1 p2p2 p3p3 ……pkpk

The probability p i must satisfy the rules of probability. We can use a probability histogram to picture the probability distribution of a discrete random variable.

The mean of a random variable x (  x ) is like a weighted average that takes into account that all the outcomes are not equally likely.  x is called the expected value. That does not mean we expect to get that value on any one trial, it means that we expect to get that average value, overall, in the long run.

Given a Discrete Probability Distribution: Value of x x1x1 x2x2 x3x3 ……xkxk Probability p1p1 p2p2 p3p3 ……pkpk

We do a weighted average to find the mean  x = x 1 p 1 + x 2 p 2 + … + x k p k =  x i p i The variance of X is = (x 1 -  x ) 2 p 1 + (x 2 -  x ) 2 p 2 + … + (x k -  x ) 2 p k =  (x i -  i ) 2 p i

The Standard Deviation (  x ) of X is the square root of the variance. Both of the previous formulas are on the AP Stat Formula Chart.

To evaluate the formulas we use the lists since they make repetitive math easy. E(x) =  x =  x i p i can be evaluated by storing the x’s in L 1 and the probabilities in L 2. Let L 3 be the products from both lists. Now Sum L 3.

Var (x) =  (x i -  i ) 2 p i can also be evaluated in the lists. In L 4, enter (L 1 -  x ) 2 L 2 remember  x is the answer you got when you summed L3 Now Sum L 4

Rules for Means of Distributions: 1) If X is a random variable and you wish to transform the data with an equation, you may apply the same equation you would use on the data to the old mean to create a new mean. 2) If X and Y are random variables and you wish to combine or take the difference of their distributions to create a new distribution, then you either combine or take the difference (respectively) of their means for a new distribution.

Rules for Variances of Distributions: 1) If X is a random variable and you wish to transform the data with an equation, a) disregard any constant values in the equation as they will not affect the variance b) to get the variance of the new distribution, multiply the old variance by any coefficient squared. 2) If X and Y are independent random variables and you wish to create a new distribution by either adding or subtracting the old distributions, then you may add (for sums & differences) the old variances. Remember: Variances of Random variables, add; standard deviations do not.