Transcript 8.1.3 - GEOCITIES.ws

Simulations with Binomials
Mean and S.D. of Binomials
Section 8.1.3
Starter 8.1.3
• 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.
2.
3.
4.
5.
•
State the problem
State the assumptions
Assign digits to outcomes
Perform the simulation many times
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:
µx = Σxipi
 X2   ( xi   x )2 pi
• In the special case of a binomial random
variable, the formulas are much simpler:
 X  np
 X2  np (1  p )
– Note: (1-p) is often written as q for simplicity
– Also: We often speak of standard deviation, so:
 X  npq
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) = 1.054
• 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