Discrete Distributions Chapter 7 Discrete Distributions

Random Variable Numerical variable whose value depends on chance Capital letters – X or Y

Two types: Discrete: X = a count of some random variable Continuous: X = a measurement of some random variable

Discrete Probability Distribution Gives probabilities for each possible X value Displayed in a table, histogram, or formula

Discrete Probability Distributions For every possible x value, 0 < P(x) < 1 For all values of x, Σ P(x) = 1

Suppose you toss 3 coins & record the number of heads. The random variable X would be… Create a probability distribution. Draw the histogram for this distribution. 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 .39 P(X > 5) = .61

Formulas for Mean & Variance Found on green sheet!

Suppose you toss 3 coins & record the number of heads. Find the mean and standard deviation for the number of heads out of 3 tosses. x 0 1 2 3 P(x) .125 .375 .375 .125 μ = 1.5 & σ = .866

What is the mean and standard deviation 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 deviation of this distribution? μ = 4.66 & σ = 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 profit! x 0 5 20 P(x) 7/9 1/6 1/18 No – μ = $1.944, which is less than the cost of playing.

Linear Transformation of a Random Variable The mean is changed by addition & multiplication The standard deviation is ONLY changed by multiplication If X and Y are random variables, and Y = a + bX, then:

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 are 318 gal and 42 gal, respectively. The company is considering 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? μ = $622.40 & σ = $75.60

Linear Combinations of Random Variables Just add or subtract the means If: Then: If the variables are independent, always add the variances

A nationwide standardized test 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 total test score? μ = 68 & σ = 9.2195