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# Distribution of Functions with Random Variables

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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.

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## 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

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

The marginal distribution for the random variable Y is

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

From the graph, find .

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.

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

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?

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?

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

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?

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

What is the expected value of X?

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.