ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 48 - 54 of 165

This proof is a lack of memory property. We known that

Therefore ………… (1)

The equation is

Using equation (1)

Since the equation is true. Therefore,

The seven ordinates of the integrand using Simpson’s rule is

Where h = 1/6 (b-a) and y _k =f (a + kh) for k = 0,1, 2,3, 4,5, 6

Here, b = 1, a = 0, h = 1/6, f (x) =

Using Simpson’s r

For find the median and quartile of the Cauchy distribution, q is any quartile. Then, for which value of q, the x value is

For first quartile q = 1/4, then x=-1

For median q = 1/2, then x = 0

For third quartile q = 3/4, then x = 1

The Euler-Maclaurin formula is

where

Given the equation

Let X ₁, X ₂, …, X _n, be n independently identically distributed random variables following an exponential distribution with mean parameter θ. The probability density and distribution function is

Let X ₍₁₎ , X ₍₂₎ , …, X _(n) denote the order statistic of a random sample, X ₁, X ₂, …, X _n from ex

The characteristic function of X is

Let assume x-µ=v, then dx = dv

Assume that , then dv= λ du

First find the value of this integral, this solution adopts the method of contour integration.

Let assume t > 0, For

We known that in continuous random variable, the mean is

We know that F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

Hence it’s proofed.