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Chapter 5: Q. 5.20 (page 216)

Verify thatVar(X)=伪位2whenXis a gamma random variable with parameters 伪and 位

Short Answer

Expert verified

To get the Variance, find the generic expression for EXkand use it.

Step by step solution

01

Determine the positive integral. 

Let's find the first and second moments of Xif X~Gamma (伪,位)

For any positive integerk.

we have that

localid="1649619459546" EXk=鈭�0鈭�xk路位伪螕(伪)x伪-1e-位xdx=位伪螕(伪)鈭�0鈭�xk+伪-1e-位xdx

02

Expression the integral.

in order to evaluate the integral, make the substitutions=位x

Hence, the expression above is equal to localid="1649619493912" 位伪螕(伪)鈭�0鈭�xk+伪-1e-位xdx=位伪螕(伪)鈭�0鈭�s位k+伪-1e-sds位

localid="1649619476293" =1位k路螕(伪)鈭�0鈭�sk+伪-1e-sds=螕(k+伪)位k路螕(伪)

So, we have obtained that,

EXk=螕(k+伪)位k路螕(伪)

03

Implies the value.

The implies that

E(X)=螕(伪+1)位脳螕(伪)=伪脳螕(伪)位脳螕(伪)=伪位

and

EX2=螕(伪+2)位2脳螕(伪)=(伪+1)伪螕(伪)位2脳螕(伪)=(伪+1)伪位2

Finally, we have that

Var(X)=EX2-E(X)2=(伪+1)伪位2-伪2位2=伪位2

Which has to be proved.

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Most popular questions from this chapter

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.

(a) Show that E[g'(Z)]=E[Zg(Z)];

(b) Show that E[Zn+1]=nE[Zn-1].

(c) Find E[Z4].

Let Xbe a random variable with probability density function

f(x)=c(1-x2)鈭�1<x<10otherwise

(a) What is the value of c?

(b) What is the cumulative distribution function of X?

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function

f(x)=5(1鈭�x)40<x<10otherwise

what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is role="math" localid="1646634562935" .01?

To be a winner in a certain game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on (0,1). If U>.1, then you are successful in round 1; if U>1, then you are successful in round 2; and if U>3, then you are successful in round 3.

(a) Find the probability that you are successful in round 1.

(b) Find the conditional probability that you are successful in round 2given that you were successful in round 1.

(c) Find the conditional probability that you are successful in round 3given that you were successful in rounds 1and2

(d) Find the probability that you are a winner

Show that 螕12=蟺

Hint: 螕12=鈭�0鈭�e-xx-1/2dxMake the change of variables y=2xand then relate the resulting expression to the normal distribution.
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