Discrete Distributions

Random Variable - A numerical variable whose value depends on the outcome of a chance experiment

Two types: Discrete – count of some random variable Continuous – measure of some random variable

Discrete Probability Distribution Gives the values associated with each possible x value Usually displayed in a table, but can be displayed with a histogram or formula

Properties for a discrete probability distribution For every possible x value, 0 < P(x) < 1. 2) For all values of x, S P(x) = 1.

Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... Create a probability distribution. Create a probability histogram. The number of heads tossed X 0 1 2 3 P(X) .125 .375 .375 .125

Why does this not start at zero? Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05 P(x = 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? Why does this not start at zero? .25 .14 P(x > 5) = .61 .39

Formulas for mean & variance Found on formula card!

What is the mean and standard deviations of this distribution? Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 .25 .40 .16 .05 What is the mean and standard deviations of this distribution? m = 4.66 & s = 1.2018

Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? A fair game is one where the cost to play EQUALS the expected value! X 0 5 20 P(X) 7/9 1/6 1/18 NO, since m = $1.944 which is less than it cost to play ($3).

Linear function of a random variable The mean is changed by addition & multiplication! If x is a random variable and a and b are numerical constants, then the random variable y is defined by and The standard deviation is ONLY changed by multiplication!

Let x be the number of gallons required to fill a propane tank Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? m = $622.40 & s = $75.60

Linear combinations Just add or subtract the means! If independent, always add the variances!

A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be mean SD MC 38 6 FR 30 7 If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test? m = 68 & s = 9.2195