In Exercise 6.4, we considered a random variable Y that

Chapter 6, Problem 24E

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QUESTION:

Problem 24E

In Exercise 6.4, we considered a random variable Y that possessed an exponential distribution with mean 4 and used the method of distribution functions to derive the density function for U = 3Y + 1. Use the method of transformations to derive the density function for U .

Reference

The amount of flour used per day by a bakery is a random variable Y that has an exponential distribution with mean equal to 4 tons. The cost of the flour is proportional to U = 3Y + 1.

a Find the probability density function for U .

b Use the answer in part (a) to find E(U ).

Questions & Answers

QUESTION:

Problem 24E

In Exercise 6.4, we considered a random variable Y that possessed an exponential distribution with mean 4 and used the method of distribution functions to derive the density function for U = 3Y + 1. Use the method of transformations to derive the density function for U .

Reference

The amount of flour used per day by a bakery is a random variable Y that has an exponential distribution with mean equal to 4 tons. The cost of the flour is proportional to U = 3Y + 1.

a Find the probability density function for U .

b Use the answer in part (a) to find E(U ).

ANSWER:

Step 1 of 2

a)The probability density function for U is expressed as:
FU(u) = Pr(U ? u) = Pr(3Y+1 ? u)
By transforming the random variable, we have:
Y ? (u-1) / 3
Therefore, the probability density function for U is:
fU(u) = fY((u-1) / 3) * (3 / (1/4)) = 12fY((u-1) / 3) = 12e^-((u-1) / 3) / 4

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