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Chapter 6: Q.6.49 (page 274)

Let X(1),X(2),...,X(n)be the order statistics of a set of n independent uniform (0,1)random variables. Find the conditional distribution of X(n) given thatX(1)=S1,X(2)=S2,....,X(n-1)=S(n-1).

Short Answer

Expert verified

Conditional distribution isX(n)=f(xn0<xn-1<1)

Step by step solution

01

Conditional distribution :

If X and Y are two jointly distributed random variables, the conditional distribution of Y given X is the probability distribution of Y when X has a known value.

02

Explanation : 

Let,

fxnx1....x(n-1)=f(xn,f(x1),...(xn-1))f(x1)....f(xn-1)

By definition of x1andx2.

fxnx1....x(n-1)=f(xn)fxnx1....x(n-1)=f(xn)....0<x(n)<1fxnx1....x(n-1)=f(xn)...x(n-1)fxnx1....x(n-1)=f(xn0<xn-1<1)

A uniform distribution has a partial differential function and a continuous differential function.

f(x)=1:0<x<1f(x)=x:0<x<1[f(xn)=n[f(x)n-1.f(x)]]f(x(n))=n[x]n-1:0<x<1f(x(n)/x(1).....x(n-1)):nxn-1:0<x<1

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Most popular questions from this chapter

Suppose that X, Y, and Z are independent random variables that are each equally likely to be either 1 or 2. Find the probability mass function of

(a) XYZ,

(b) XY+XZ+YZ, and

(c)X2+YZ

The joint density of X and Y is

f(x,y)=cx2-y2e-x0鈮�x<鈭�,-x鈮�y鈮�x

Find the conditional distribution of Y, given X = x.

If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function fX+Y(t)=q鈭�qfX,Y(x,t鈭�x)dx

Let Z1,Z2......Znbe independent standard normal random variables, and let Sj=鈭�i=1jZi

(a) What is the conditional distribution of Sn given that Sk=y,for k = 1, ... , n?

(b) Show that, for 1 鈥� k 鈥� n, the conditional distribution of SKgiven that

Sn = x is normal with mean xk/n and variance k(n 鈭� k)/n.

In Problem 6.5, calculate the conditional probability mass function of Y1 given that

(a) Y2=1

(b)Y2=0
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