Probability(C14-C17 BVD) C16: Random Variables

* 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).

* 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) = Σx i p i

* Variance = σ 2 = Σ(x i -µ) 2 p i * ALWAYS do calculations using variances, then take square root at end to get σ * Calculator

* 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).

* If Y = a +bX is a transformation of a random variable X, then… * µ y = a + bµ x * The mean or Expected value of Y is just the mean of the original distribution times b added to a. * σ y 2 = b 2 σ x 2 * The variance of Y is the variance of X times b 2. The “a” does NOT affect spread.

* 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) * µ y = a + bµ x = /5(34) = 93.2 degrees Fahrenheit * σ y 2 = b 2 σ x 2 = 81/25(4) = * => σ y = 3.6 degrees Fahrenheit

* 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.

* 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! * µ y = a + bµ x vs. E(X+Y) = E(X) + E(Y) * 2(0.25) vs => 0.5 vs. 0.5 * σ y 2 = b 2 σ x 2 vs. Var(X+Y) = Var(X) + Var(Y) * 2 2 (0.1) 2 vs. (0.1) 2 + (0.1) 2 => 0.04 vs * => 0.2 vs for standard deviations See pages for examples