a. Let X1 have a chi-squared distribution with parameter v1 (see Section 4.4), and let X...

a. Let X₁ have a chi-squared distribution with parameter v1 (see Section 4.4), and let X₂ be independent of X₁ and have a chi-squared distribution with parameter v₂. Use the technique of Example 5.21 to show that X₁ + X₂ has a chi-squared distribution with parameter v₁ _ v₂.

b. In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Z² has a chi-squared distribution with v = 1. Let Z₁, Z ₂, . . . , Z_n be n independent standard normal rv’s. What is the distribution of ? Justify your answer.

c. Let X₁, . . . , X_n be a random sample from a normal distribution with mean m and variance σ². What is the distribution of the sum ? Justify your answer.

Reference example 5.21

Service time for a certain type of bank transaction is a random variable having an exponential distribution with parameter l. Suppose X1 and X2 are service times for two different customers, assumed independent of each other. Consider the total service time To = X1 + X2 for the two customers, also a statistic. The cdf of T_o is, for t ≥ 0,

Reference exercise 7 in chapter 4

The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with A = 25 and B =35.

a. Determine the pdf of X and sketch the corresponding density curve.

b. What is the probability that preparation time exceeds 33 min?

c. What is the probability that preparation time is within 2 min of the mean time? [Hint: Identify μ from the graph of f(x).]

d. For any a such that 25 < a < a + 2 < 35, what is the probability that preparation time is between a and a+2 min?