Simulations with Binomials Mean and S.D. of Binomials Section 8.1.3

Starter A baseball player bats.325 over a full season. If he bats 5 times today, find the probability he gets at least 3 hits.

Objectives Approximate the PDF of a binomial random variable by performing a simulation Calculate the mean and standard deviation of a binomial random variable from formulas California Standards 5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable. 6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable. 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

Simulate a Binomial Random Variable Recall the 5 steps involved in a simulation of a random variable: 1.State the problem 2.State the assumptions 3.Assign digits to outcomes 4.Perform the simulation many times 5.State your conclusions Run one simulation of the dice game from yesterday and report how many wins you get in 5 rolls. (We will combine class results to get many simulations and draw conclusions). –Don’t shortcut the process: Actually write the first three steps on paper before doing the simulation.

Answer Steps 1 – 3 1.Problem: Estimate the PDF of the binomial random variable in the dice game 2.Assumptions: Rolls are independent and p(success) is fixed at 1/3, so binomial setting applies 3.Use digits 1 – 6 and assign 1 & 2 to mean win; assign 3 – 6 to mean lose OR: Use digits 1 – 3 with 1 meaning win; 2 & 3 lose Tell me how many wins you got in 5 rolls. Compare simulated PDF with theoretical

Mean and Variance Recall the moderately complicated formulas for mean and variance of any random variable: µ x = Σx i p i In the special case of a binomial random variable, the formulas are much simpler: –Note: (1-p) is often written as q for simplicity –Also: We often speak of standard deviation, so:

Example Calculate the mean and standard deviation of the dice game based on the formulas –μ = 5 x (1/3) = 5/3 = 1.67 So we expect to win 1.67 times per set of 5 games –σ = √(5)(1/3)(2/3) = Compare those answers to the results you get if you put the PDF into lists and run the 1-var stats command.

Objectives Approximate the PDF of a binomial random variable by performing a simulation Calculate the mean and standard deviation of a binomial random variable from formulas California Standards 5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable. 6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable. 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

Homework Read pages 428 – 430 Do problems 11 & 15