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 that $X_{(1) = S_{1}}, X_{(2) = S_{2}}, . . . ., X_{(n - 1) = S (n - 1)}$ .

Short Answer

Expert verified

Conditional distribution is $X_{(n)} = f (x_{n} 0 < x_{n - 1} < 1)$

Step by step solution

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.

Explanation :

Let,

$f (\frac{x_{n}}{x_{1} . . . . x_{(n - 1)}}) = \frac{f (x_{n}, f (x_{1}), . . . (x_{n - 1}))}{f (x_{1}) . . . . f (x_{n - 1})}$

By definition of $x_{1} a n d x_{2}$ .

$f (\frac{x_{n}}{x_{1} . . . . x_{(n - 1)}}) = f (x_{n}) f (\frac{x_{n}}{x_{1} . . . . x_{(n - 1)}}) = f (x_{n}) . . . . 0 < x_{(n)} < 1 f (\frac{x_{n}}{x_{1} . . . . x_{(n - 1)}}) = f (x_{n}) . . . x_{(n - 1)} f (\frac{x_{n}}{x_{1} . . . . x_{(n - 1)}}) = f (x_{n} 0 < x_{n - 1} < 1)$

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

$f (x) = 1 : 0 < x < 1 f (x) = x : 0 < x < 1 [f (x_{n}) = n [f {(x)}^{n - 1} . f (x)]] f (x_{(n)}) = n {[x]}^{n - 1} : 0 < x < 1 f (x_{(n)} / x_{(1)} . . . . . x_{(n - 1)}) : n x^{n - 1} : 0 < x < 1$

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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 that $X_{(1) = S_{1}}, X_{(2) = S_{2}}, . . . ., X_{(n - 1) = S (n - 1)}$ .

Short Answer

Step by step solution

Conditional distribution :

Explanation :

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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

Step by step solution

Conditional distribution :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

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 that $X_{(1) = S_{1}}, X_{(2) = S_{2}}, . . . ., X_{(n - 1) = S (n - 1)}$ .