Download Probability Distributions Practice/Review

2016-2017 AP Statistics
Probability Distributions Review
Name ______________________________
Purpose: To review the four (4) probability distributions we have studied this year.
Definitions:
The probability distribution of a discrete random variable ___________________________________________________
__________________________________________________________________________________________________
__________________________________________________________________________________________________
The probability distribution of a continuous random variable is described ______________________________________
__________________________________________________________________________________________________
__________________________________________________________________________________________________
Distribution #1
The Uniform Distribution
--Each outcome is equally likely to occur.
--Uniform distributions can describe discrete random variables (rolling a die) or continuous random variables (select
a random number between 0 and 1)
--The probability histogram or density curve of a uniform distribution is rectangular in shape, with the values of the
random variable on the x-axis and the probability of each occurring on the y-axis.
--For continuous random variables, the area under the curve must be 1. Therefore, the height of a uniform
1
distribution that takes the values [a, b] must be 𝑏−𝑎.
Example: Discrete Random Variable with a Uniform Distribution.
A game has a spinner with 5 sections, each numbered either 1, 2, 3, 4, or 5 (no repeats). The number you spin is how far
you get to move your game piece. (a) Sketch the probability histogram for the spinner and (b) fill in the table with the
probability distribution of the spinner.
X
P(X)
Example: Continuous Random Variable with a Uniform Distribution
Buses arrive at a given bus stop every 15 minutes. Sketch the probability distribution of the time you will need to wait
for a bus if you arrive at a random time.
(a) Calculate 𝑃(𝑋 < 5) and describe in words what it means.
(b) Calculate 𝑃(𝑋 ≥ 10) and describe in words what it means.
(c) Calculate 𝑃(7 < 𝑋 < 12) and describe in words what it means.
(d) Calculate 𝑃(𝑋 ≤ 5).
(e) Explain what your answers to parts (a) and (d) are the same.
Distribution #2
The Normal Distribution
--The normal distribution is a continuous random variable whose distribution is a bell shaped curve.
--The height and width of the curve (its shape) is determined by the mean (𝜇) and standard deviation (𝜎) of the
random variable.
--The area under the curve represents probabilities of events happening.
--The standard normal curve is a normal distribution with mean 𝜇 = 0 and standard deviation 𝜎 = 1. All random
𝑥−𝜇
variables that have a normal distribution can be standardized using the z-score (𝑧 = 𝜎 ). NOTE: ALL variables can
be standardized using a z-score, but the probabilities calculated using the Normal curve have no meaning if the
original variable does not follow a normal distribution
--Probabilities may be calculated using a standard normal table or the normalcdf command on the calculator
--normalcdf(lower bound, upper bound, mean, standard deviation)
Example: Normal Distribution
Suppose that the weight of navel oranges is normally distributed with mean 𝜇 = 8 ounces, and standard deviation 𝜎 =
1.5 ounces.
(a) What is the probability a randomly selected orange weighs less than 8.7 ounces?
(b) What is the probability a randomly selected orange weighs more than 11.5 ounces?
(c) What is the probability a randomly selected orange weighs between 6.2 and 7 ounces?
(d) What is the probability a randomly selected orange weighs between 10.3 and 14 ounces?
Distribution #3
The Binomial Distribution
--The binomial distribution describes a discrete random variable that meets the following requirements:
--There are only two outcomes that can be defined as “success” or “failure”
--The probability of success is the same on each outcome (p)
--Each trial is independent
--The values of the variable (X) are the number of success in a fixed number of trials (n)
--To calculate the probability of some number of successes in n trials,
𝑛
𝑃(𝑋 = 𝑟) = ( ) 𝑝𝑟 (1 − 𝑝)𝑛−𝑟
𝑟
where p is the probability of success on each trial and r is the number of successes
--The calculator may be used to calculate probabilities
--𝑃(𝑋 = 𝑟) = 𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓(𝑛, 𝑝, 𝑟) and 𝑃(𝑋 ≤ 𝑟) = 𝑏𝑖𝑛𝑜𝑚𝑐𝑑𝑓(𝑛, 𝑝, 𝑟)
--The mean and standard deviation of a binomial distribution can be calculated as
𝜇 = 𝑛𝑝
and
𝜎 = √𝑛𝑝(1 − 𝑝)
--Binomial probabilities can be approximated using 𝑁(𝑛𝑝, √𝑛𝑝(1 − 𝑝)) if 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10
Example: Binomial Distribution
A company owns 400 laptops. Each laptop has an 8% probability of not working. You randomly select 20 laptops for
your salespeople.
(a) What is the probability that 5 will be broken?
(b) What is the probability that they will all work?
(c) What is the probability you will need to find at least 4 more laptops?
(d) What is the probability that more than half the laptops are broken?
(e) What is the probability at most 5 laptops are broken?
Distribution #4
The Geometric Distribution
--The geometric distribution describes a discrete random variable that meets the following requirements:
--There are only two outcomes that can be defined as “success” or “failure”
--The probability of success is the same on each outcome (p)
--Each trial is independent
--The values of the variable (X) are the number of trials needed to achieve the first success
--To calculate the probability of success happening on the nth trial,
𝑃(𝑋 = 𝑛) = (1 − 𝑝)𝑛−1 𝑝
where p is the probability of success on each trial and r is the number of successes
--The calculator may be used to calculate probabilities
--𝑃(𝑋 = 𝑟) = 𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓(𝑝, 𝑛) and 𝑃(𝑋 ≤ 𝑛) = 𝑔𝑒𝑜𝑚𝑒𝑡𝑐𝑑𝑓(𝑝, 𝑛)
--The mean and standard deviation of a binomial distribution can be calculated as
1
𝜇=𝑝
and
(1−𝑝)
𝑝2
𝜎=√
--The calculation 𝑃(𝑋 > 𝑛) = (1 − 𝑝)𝑛 , because the fact that n failure has occurred is the only known fact.
Example: Geometric Distribution
Megan is randomly calling businesses in her town (all she has is a list of phone numbers) searching for a restaurant that
offers delivery. 15% of the business in her town are restaurants that offer delivery, and assume independent trials.
(a) Calculate 𝑃(𝑋 = 5) and explain in words what it represents.
(b) Calculate 𝑃(𝑋 ≤ 10) and explain in words what it represents.
(c) Calculate 𝑃(𝑋 > 10) and explain in words what it represents.
(d) How many phone numbers can Megan expect to call before she finds a restaurant that offers delivery?
(e) Calculate the standard deviation and explain in words what it represents.
Calculating Expected Value and Standard Deviation of Other Probability Models
Discrete Probability Models
--To calculate the mean and variance of a discrete probability model,
𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑝(𝑥)
𝜎 2 = 𝑉𝑎𝑟(𝑋) = ∑(𝑥 − 𝜇)2 𝑝(𝑥)
--The sum of two random variables changes the mean and standard deviation so that
𝜇𝑥+𝑦 = 𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌)
2
𝜎𝑥±𝑦 = 𝑉𝑎𝑟(𝑋 ± 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌), if X and Y are independent
--The transformation of a random variable changes the mean and standard deviation so that
𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑥
2
𝜎𝑎+𝑏𝑋
= 𝑏 2 𝜎𝑋2
In a recent little league softball game, each player went to bat 4 times. The number of hits made by each player is
described by the following probability distribution.
Number of Hits, X
0
1
2
3
4
P(X)
0.10
0.20
0.30
0.25
0.15
(a) Calculate the mean and standard deviation of the number of hits each player made.
(b) In a second game, each player only went to bat 3 times. The number of hits made by each player is described by the
following probability distribution.
Number of Hits, X
0
1
2
3
P(X)
0.10
0.30
0.35
0.25
Calculate the mean and standard deviation of the number of hits each player made.
(c) Calculate the mean and standard deviation of the number of hits made by each player in both games.
(d) In the second game, every player got one more hit than was recorded (so everyone made at least 1 hit). Calculate
the new mean and standard deviation for the second game (the probabilities did not change)
(e) Due to extra innings, each player got twice as many hits in the first game that what was recorded and three times as
many hits in the second game! Calculate the new mean and standard deviation for the number of hits made by each
player in both games.