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Chapter 6: Q.6.27 (page 277)

Establish Equation (6.2)by differentiating Equation6.4.

Short Answer

Expert verified

On differentiating equation(6.4)we get equation(6.2).

Step by step solution

01

Differentiation 

Differentiation is a technique for calculating a derivative, which is the rate of change of a function's output yin relation to the change of a variable x.

02

Explanation : 

Fxij(y)=∑k=jnnkF(y)k1-F(y)n-k..........(6.4)

Differentiating both sides with respect to y, then

fXj(y)=∑k=jnnkkF(y)k-1f(y)[1-F(y)]n-k-∑k=jnnkF(y)k-1(n-k)[1-F(y)]n-k-1f(y)=∑i=jnn!(n-k)!(k-1)!F(y)k-1f(y)[1-F(y)]n-k-∑k=j+1nn!(n-k)!(k-1)!Fk-1(y)f(y)[1-F(y)]n-kbyj=k+1=n!(n-j)!(j-1)!Fj-1(y)f(y)[1-F(y)]n-j

which is the same as the equation (6.2)

Hence proved.

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Most popular questions from this chapter

Let X1,X2,...be a sequence of independent uniform (0,1)random variables. For a fixed constant c, define the random variable N by N=min{n:Xn>c}Is N independent ofXN? That is, does knowing the value of the first random variable that is greater than c affect the probability distribution of when this random variable occurs? Give an intuitive explanation for your answer.

If X, Y, and Z are independent random variables having identical density functions f(x)=e-x,0<x<∞ derive the joint distribution of U=X+Y,V=X+Z,W=Y+Z.

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}

If U is uniform on (0,2π)and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined by

X=2ZcosUY=2ZsinU

are independent standard normal random variables.

Consider independent trials, each of which results in outcome i, i = 0,1,....,k, with probability pi,∑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≥1

(b) Find P{X=j},j=1,…,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?
See all solutions

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