Download Random Variables and Combining Random Variables WS

Combining and Transforming Random Variables Worksheet
Name: _____________________________________________________ Date: ____________ Period: _____
Linear Transformations of Random Variables
If Y = a + bX is a linear transformation of the random variable X, then:
• The probability distribution of Y has the same shape as the probability distribution of X.
• 𝜇𝑌 = 𝑎 + 𝑏𝜇𝑋
• 𝜎𝑌 = |𝑏|𝜎𝑥 (since b could be a negative number).
• 𝜎𝑌2 = 𝑏 2 𝜎𝑋2
1) In a large introductory statistics class, the distribution of X = raw scores on a test was
approximately normally distributed with a mean of 17.2 and a standard deviation of 3.8. The
professor decides to scale the scores by multiplying the raw scores by 4 and adding 10.
a. Define the variable Y to be the scaled score of a randomly selected student from this
class. Find the mean and standard deviation of Y. Show your work.
b. What is the probability that a randomly selected student has a scaled test score of at
least 90? Show your work.
2) A small ferry runs every half hour from one side of a large river to the other. The number of
cars X on a randomly chosen ferry trip has the probability distribution shown below. You can
that 𝜇𝑋 = 3.87 and 𝜎𝑋 = 1.29
Cars
Probability
0
0.02
1
0.05
2
0.08
3
0.16
4
0.27
5
0.42
a) The cost for the ferry trip is $5 per car. Make a graph of the probability distribution for the
random variable M = money collected on a randomly selected ferry trip.
b) Find 𝜇𝑀 and 𝜎𝑀 and interpret.
c) The ferry expenses are $20 per trip. Define random variable Y to be the amount or profit
(money collected minus expenses) made by the ferry company on a randomly selected day.
Find 𝜇𝑌 and 𝜎𝑌 . Interpret the mean. Show your work.
d) The ferry company now decides to increase the cost of a trip to $6. We can calculate the
company’s profit Y on a randomly selected trip from the number of cars X. Find the mean
and standard deviation of Y. Show your work.
Combining Random Variables
Suppose I am offered a chance to play a game involving two spinners. Complete the spinners to show the
given probability distributions:
Spinner X
X
P(X)
2
0.3
Spinner Y
4
0.4
5
0.1
8
0.2
X
P(X)
3
0.2
4
0.2
5
0.6
1) What is the expected value, variance and standard deviation for each spinner?
𝜇𝑥 = _______
Var(X) = ______
𝜎𝑥 = ______
𝜇𝑦 = _______
Var(Y) = ______
𝜎𝑦 = ______
2) In this game, you will each of the spinners once. What are the possible outcomes and probabilities for
the sum of the two spinners? Fill in the following table to form the probability distribution of X + Y.
The whole chart should be completed.
(X, Y)
(2, 3)
(2, 4)
X+Y
5
6
P(X + Y)
0.06
0.06
3) What is the expected value, variance, and standard deviation for the game as described by your answer
in part 2?
𝜇𝑋+𝑌 = ___________
Var(X + Y) = _________
𝜎𝑋+𝑌 = ___________
4) Suppose that the game is changed to represent the different between the two spinners. Thus your
return on any game is X – Y. Complete the table below to find the probabilities of the difference of the
two spinners.
(X, Y)
X-Y
P(X - Y)
5) What is the expected value, variance and standard deviation for the game described in part 4?
𝜇𝑋−𝑌 = ___________
Var(X - Y) = _________
𝜎𝑋−𝑌 = ___________
Summary: Use your data you found in #1, #3 and #5 and comparing means, variances and standard
deviations to generalize into 4 rules listed below.
Rules for Means:
For any two random variables X and Y, then
𝜇𝑋+𝑌 = _________________
For any two random variables X and Y, then
𝜇𝑋−𝑌 = _________________
Rules of Variance/Standard Deviation
For any two independent random variables X and Y, then the variance of X + Y is
2
𝜎𝑋+𝑌
= _________________ and therefore, 𝜎𝑋+𝑌 = _______________________
For any two independent random variables X and Y, then the variance of X − Y is
2
𝜎𝑋−𝑌
= _________________ and therefore, 𝜎𝑋−𝑌 = _______________________
Note: Random Variables X and Y must be INDEPENDENT in order to apply combining random variable
rules. Probability models often assume independence when the random variables describe outcomes that
appear unrelated to each other.