Download Transforming and Combining Random Variables

AP Statistics
Name: _____________________
Lesson 6.3 Transforming and Combining Random Variables
Effect of Linear Transformations on a Random Variable
Ex. #1 A small ferry runs every half hour from one side of large river to the other. The number of cars X on a
randomly chosen ferry trip has the probability distribution shown below. Please use calculator for work
below.
Cars
Probability
0
0.02
1
0.05
2
0.08
3
0.16
4
0.27
5
0.42
a) Find the mean and standard deviation of the random variable (X).
b) Make a graph of the probability distribution for the random variable (X). Describe its shape.
c) The cost of the ferry trip is $5. Make a graph of the probability distribution for the random variable
M=money collected on a randomly selected ferry trip. Describe its shape.
d) Find and interpret µ M and σ M .
e) The ferry’s company’s expenses are $20 per trip. Define the random variable Y to be the amount of
profit (money collected minus expenses) made by the ferry company on a randomly selected trip. Write
an equation for Y in terms of M.
f) How does the mean of Y relate the mean of M? What is the practical importance of µY ?
g) How does the standard deviation of Y relate to the standard deviation of M? What information does
σ Y give us?
Summary: Linear Transformation of a Random Variable
AP Statistics
Name: _____________________
Combining Random Variables
Rules for combining Random Variables:
Ex. #1 A college uses SAT scores as one criterion for admission. Experience has shown that the distribution of
SAT scores among its entire population of applicants is:
SAT Math Score (X)
SAT Verbal Reasoning (Y)
µ X = 625
µY = 590
σ X = 90
σ Y = 100
What are the mean and standard deviation of the combined Math and Verbal Reasoning scores amoung
the students applying to this college?
Ex. #2 An electronics store has determined that its sales of big screen TVs has the following distribution:
X= # units sold
100 200 400 700
P(X)
0.2
0.4
0.3
0.1
a) Is X discrete or continuous?
b) Calculate the mean, variance and standard deviation for the probability distribution.
c) Suppose that the store has a cost structure determined by the formula C = 55 X + 250 . Create the
probability distribution for the Cost (C). (we are transforming the random variable, X)
Cost (C)
100 200 400 700
P(C)
0.2
0.4
0.3
0.1
AP Statistics
Name: _____________________
d) Suppose the store also sells computers. The sale of computers follows the distribution below:
Y= # units sold
e) P(Y)
200 300 500 1000
0.1
0.2
0.5
0.2
E
e) Find the mean, variance and standard deviation for Y.
f) Suppose the profit is given by the formula: Z = 200 X + 350Y . Find the mean and standard deviation of
the store’s profit.
Combining Normal Random Variables:
Ex. #3 Tom and George are playing in the club golf tournament. Their scores vary as they play the course
frequently. Both men’s scores follow a normal distribution. Tom’s score (X) has a mean of 110 with a
standard deviation of 10 and George’s score (Y) has a mean of 100 with a standard deviation of 8. If
they play independently, what is the probability that Tom will score lower than George and
consequently do better in the tournament?
Homework: pg. 378 # 35, 39, 40, 49, 51, 53, 55, 57, 61, Multiple Choice #65-66 (quiz next class)