# Probability and Cumulative Distribution Function

1. A random variable X has the following cumulative distribution function

F(x) = { 1 - e^(-(x+1)) -1</ x < oo

0 elsewhere.

a) 25% of the time, X exceeds what value?

b) Find the moment generating function of X, or Mx(t)

c) Using your result in (b), find the E(X^2)

2. The lifetime of a certain brand of tire, in 10's of thousands of kilometers (or 10,000 kms), is a random variable having the following probability density function.

f(x) = { 20/(3x^2) 4</ x </ 10

0 elsewhere.

a) find the cumulative distribution function of X, or

F(x)

b) How long would you expect one of these tires to last, in kilometers?

c) If a tire lasts for at least 65,000 kilometers, what is the probability it will last at most 85,000 kilometers?

3. (refer to question 2). X1, X2, ...,Xk represent a random sample of k-tires

a) What is the probability that the 28th tire selected is the 5th tire to have a lifetime exceeding 90,000 kms?

b) To ensure that there is a probability of 0.90 that at least one tire in the k sampled will have a lifetime exceeding 90,000 kms, how many tires should be randomly selected? (k=?)

4. The number of mice in a grain bin, at any given time, is a random variable which follows a Poisson distribution with a mean of 4.6.

a) What is the probability that there is exactly 12 mice in two randomly selected grain bins?

b) What is the probability that 10 of the 50 randomly selected grain bins have exactly six mice?

#### Solution Preview

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1. A random variable X has the following cumulative distribution function

F(x) = { 1 - e^(-(x+1)) -1</ x < oo

0 elsewhere.

a) 25% of the time, X exceeds what value?

Solution. Since the cumulative distribution function is

So, the density function is

Now, we want to find x such that , so .

Let = , we have . So

b) Find the moment generating function of X, or Mx(t)

Solution. By the definition, we have

Note: when t>=1 , the expectation is not finite

c) Using your result in (b), find the E(X^2)

Solution. It is easy to know that = . Now we compute

and

So, =1,

Thus, = =1.

2. The lifetime of a certain brand of tire, in 10's of thousands of kilometers (or 10,000 kms), is a random variable having the following probability density function.

f(x) = { 20/(3x^2) 4</ x </ 10

0 elsewhere.

a) find the cumulative distribution function of X, or

F(x)

Solution. By the definition, we have

= ...

#### Solution Summary

The probability and cumulative distribution functions are analyzed. The moment generating functions of X are given. The solution answers the question(s) below.