Probability(C14-C17 BVD) C16: Random Variables. * Random Variable – a variable that takes...

Probability(C14-C17 BVD)

C16: Random Variables

*AP Statistics Review

Random Variables and Probability

Distributions

*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).

*Expected Value

*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

*Variance/Standard

Deviation of Random Variables

*Variance = σ2 = Σ(xi-µ)2pi

*ALWAYS do calculations using variances, then take square root at end to get σ

*Calculator

*Continuous Random Variables

*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).

*Transformations of Random

Variables

*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 = b2σx

2

*The variance of Y is the variance of X times b2. The “a” does NOT affect spread.

*Example

*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 = b2σx

2 = 81/25(4) = 12.96

*=> σy = 3.6 degrees Fahrenheit

*Combining Random Variables

*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.

*X1 + X2 ≠ 2X

*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 say the 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 = b2σx

2 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