# Distribution of Functions with Random Variables

A sum of $50,000 is invested at a rate R, selected from a uniform distribution on the interval (0.03, 0.07). Once R is selected, the sum is compounded instantaneously for a year so that X= 50000 e^R dollars is the amount at the end of that year.

a) Find the distribution function and the p.d.f. of X.

b) Verify that X = 50000e^R is defined correctly if the compounding is done instantaneously.

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#### Solution Summary

This solution answers questions regarding the distribution of functions with random variables.

Probability, Random Variables, Joint Density Functions, Cumulative Density Functions and Projection Graphs (12 Problems)

1. Given the joint density function for the random variables X and Y as

The marginal distribution for the random variable X is

Answer:

2. Given the joint density function for the random variables X and Y as

The marginal distribution for the random variable Y is

Answer:

3. The following represents the cumulative distribution function for a random variable X.

From the graph, find .

Answer: 0.4

4. The life span in hours for an electrical component is a random variable X with cumulative distribution function

Determine the probability density function for X.

Answer:

5. Let X be the random variable for the life in hours for a certain electronic device. The probability density function is

The expected life for a component is

Answer: 2000 hours

6. The life, X in hundred of hours, of a certain battery has the following density function

What is the average life of the battery?

Answer: 200 hours

7. The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution

What is the expected or average time between successive speeders?

Answer: 0.125 hours

8. The probability distribution of X, the number of defects per 100 yards of a fabric is given by

x 0 1 2 3 4

f(x) 0.45 0.35 0.14 0.05 0.01

The variance for X is

Answer: 0.8476

9. The following represents the projection graph for a probability distribution f(x) of a random variable X.

What is the value for the variance of X?

Answer: 1

10. The following represents the cumulative distribution function for a random variable X.

What is the expected value of X?

Answer: 2.2

11. The life span in hours for an electrical component is a random variable X with cumulative distribution function

Determine the expected life span for an electrical component.

Answer: 100

12. The life span in hours for an electrical component is a random variable X with cumulative distribution function

Determine the variance for the life span for an electrical component.

Answer: 10000

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