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Chapter 6: Q. 6.7 (page 275)

(a) If X has a gamma distribution with parameters (t, 位), what is the distribution of cX, c > 0?

(b) Show that 饾挸2n22饾潃has a gamma distribution with parameters n and 位when nis a

positive integer and 饾挸2n2is a chi-squared random variable with 2n degrees of freedom.

Short Answer

Expert verified

(a) cX~饾殴t,饾潃cand

(b) 饾挸2n22饾潃~饾湠n,饾浑

Step by step solution

01

Part (a) : Given information

X~饾洡(t,饾浑)

We need to find the distribution ofcX

02

Part (a) : Calculations

We know that the M.G.F. of gamma distribution with parametern,饾浑is

localid="1651051186085" Mx(t)=1-t饾浑-n=EetX

Therefore we have

McXt=EectX=MX(ct)鈬�MX(ct)=1-ct饾浑-n

which is nothing but the distribution function of gamma distribution with parameters t,饾潃c, i.e.

cX~饾殴t,饾潃c

03

Part (b) : Given information

Here we need to show thatis a gamma distribution with parameters n and位

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Most popular questions from this chapter

Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute

(a) P{X1 > X2|X1 > X3};

(b) P{X1 > X2|X1 < X3};

(c) P{X1 > X2|X2 > X3};

(d) P{X1 > X2|X2 < X3}

Suppose that A, B, C, are independent random variables, each being uniformly distributed over0,1.

(a) What is the joint cumulative distribution function of A, B, C?

(b) What is the probability that all of the roots of the equation AX2+Bx+C=0are real?

Consider independent trials, each of which results in outcome i, i = 0,1,....,k, with probability pi,鈭�i=0Kpi=1. Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.

(a) Find P{N=n},n鈮�1

(b) Find P{X=j},j=1,鈥�,k

(c) Show that P{N=n,X=j}=P{N=n}P{X=j}.

(d) Is it intuitive to you that N is independent of X?

(e) Is it intuitive to you that X is independent of N?

The accompanying dartboard is a square whose sides are of length 6.

The three circles are all centered at the center of the board and are of radii 1, 2, and 3, respectively. Darts landing within the circle of radius 1 score 30 points, those landing outside this circle, but within the circle of radius 2, are worth 20 points, and those landing outside the circle of radius 2, but within the circle of radius 3, are worth 10 points. Darts that do not land within the circle of radius 3 do not score any points. Assuming that each dart that you throw will, independently of what occurred on your previous throws, land on a point uniformly distributed in the square, find the probabilities of the accompanying events:

(a) You score 20 on a throw of the dart.

(b) You score at least 20 on a throw of the dart.

(c) You score 0 on a throw of the dart.

(d) The expected value of your score on a throw of the dart.

(e) Both of your first two throws score at least 10.

(f) Your total score after two throws is 30.

Each throw of an unfair die lands on each of the odd numbers 1,3,5with probability C and on each of the even numbers with probability 2C.

(a) Find C.

(b) Suppose that the die is tossed. Let X equal 1if the result is an even number, and let it be 0otherwise. Also, let Y equal 1if the result is a number greater than three and let it be 0otherwise. Find the joint probability mass function of X and Y. Suppose now that 12independent tosses of the die are made.

(c) Find the probability that each of the six outcomes occurs exactly twice.

(d) Find the probability that 4of the outcomes are either one or two, 4are either three or four, and 4are either five or six.

(e) Find the probability that at least 8of the tosses land on even numbers.
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