Chapter 7 Discrete Distributions Random Variable Numerical variable

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… The number of heads tossed Create a probability distribution. x P(x) 0. 125 1. 375 2. 375 3. 125 Draw the histogram for this distribution.

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

Formulas for Mean & Variance Found on green sheet!

Suppose you toss 3 coins & record the number of heads. x P(x) 0. 125 1. 375 2. 375 3. 125 Find the mean and standard deviation for the number of heads out of 3 tosses. μ = 1. 5 & σ =. 866

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, hewhere winsthe A fair game is one cost to play EQUALS the $20. If he gets a expected 7, he wins $5. profit! Thex cost to 0 roll the 5 dice one 20 time is $3. Is this game fair? P(x) 7/9 1/6 1/18 No – μ = $1. 944, which is less than the cost of playing.

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

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