Understanding Probability Distributions with Random Variables

1. AP Statistics Review Probability(C14-C17 BVD) C16: Random Variables

2. Random Variable – a variable that takes numerical values describing the outcome of a random process. • Probability Distribution (a.k.a. probability model) – A table that lists all the outcomes a random variable can take (sample space) and the associated probabilities for each outcome. Probabilities must add to 1. • Random variable notation: X (capital) for the variable, x (often with a subscript) for an individual outcome • P(X=x) is the probability the variable takes on the value x (or that outcome x happens). Random Variables and Probability Distributions

3. Discrete Random Variable – the probability distribution is finite – you can list all the possible outcomes. • The mean of a discrete random variable is called “Expected value”, because it represents the long-run average, or what you would expect in the long-run. • µ = E(x) = Σxipi Expected Value

4. Variance = σ2 = Σ(xi-µ)2pi • ALWAYS do calculations using variances, then take square root at end to get σ • Calculator Variance/Standard Deviation of Random Variables

5. These can take all values in an interval – there an infinite number of possible outcomes. • Their distributions are density curves such as the normal model. • If a normal model is appropriate to describe the distribution, then you can use z-scores and z-table to find areas under the curve to represent probabilities (see Ch 6). Continuous Random Variables

6. If Y = a +bX is a transformation of a random variable X, then… • µy = a + bµx • The mean or Expected value of Y is just the mean of the original distribution times b added to a. • σy2 = b2σx2 • The variance of Y is the variance of X times b2. The “a” does NOT affect spread. Transformations of Random Variables

7. Temperature in a dial-set temperature-controlled bathtub for babies (X) has a mean temperature of 34 degrees Celsius with a standard deviation of 2 degrees Celsius. • Convert the mean and standard deviation to Fahrenheit degrees (F = 9/5C +32) • µy = a + bµx = 32 + 9/5(34) = 93.2 degrees Fahrenheit • σy2 = b2σx2 = 81/25(4) = 12.96 • => σy = 3.6 degrees Fahrenheit Example

8. When adding/subtracting two different random variables X and Y: • E(X+Y) = E(X) + E(Y) • E(X-Y) = E(X) – E(Y) • Var(X+Y) = Var(X) + Var(Y) • Var(X-Y) = Var(X) +Var(Y) • Notice! Variances add even when random variables are being subtracted. • Remember! Take the square root at the very end to find standard deviations. Combining Random Variables

9. Let’s say X is a random variable for the amount of a bet. X + X may represent two one dollar bets. 2X may represent a single two dollar bet. Let’s saythe expected winnings from one 1-dollar bet is $-0.25 with a standard deviation of $0.10. • Expected values may come out the same either way, but variances/standard deviations probably won’t! • µy = a + bµx vs. E(X+Y) = E(X) + E(Y) • 2(0.25) vs. 0.25 + 0.25 => 0.5 vs. 0.5 • σy2 = b2σx2 vs. Var(X+Y) = Var(X) + Var(Y) • 22(0.1)2 vs. (0.1)2 + (0.1)2 => 0.04 vs. 0.02 • => 0.2 vs. 0.141 for standard deviations See pages 315-318 for examples X1 + X2 ≠ 2X