Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 20 - 25 of 72

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Question number: 20

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X have a geometric distribution, then for an two non-negative integers m and n,

P(X>m+n X>m)=P(Xn)

Prove it

Explanation

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

P(Xn)=1qn+1

Therefore P(Xn)=qn+1 ………… (1)

The equation is

P(X>m+nX>… (154 more words) …

Question number: 21

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Obtain the median and the quartiles of the Cauchy distribution with p. d. f.

f(x)=1π(1+x2);<x<

Explanation

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

P[X<x]=xf(t)dt=q

1πx1… (84 more words) …

Question number: 22

» Probability » Characteristic Function

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Obtain the characteristic function of X whose pdf is

f(x)=λπ.1λ2+(xμ)2,<x<

Explanation

The characteristic function of X is

E(eitx)=λπeitxλ2+(xμ)2dx

Let assume x-µ = v, then dx = dv

=λπ… (403 more words) …

Question number: 23

» Probability » Expectation

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

E(X)=0(1F(x))dx0F(x)dx

Explanation

We known that in continuous random variable, the mean is

E(X)=xf(x)dx

=0xf(x)dx+0xf(x)dx

We… (135 more words) …

Question number: 24

» Probability » Expectation

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Let X 1, X 2, ···, X n be random variables such that

E(X1)=μ,E(Xr Xr1)=Xr1;r=2(1)n

Also

E(X1μ)2=σ2,E[(XrXr1)2 Xr1]=σ2;r=2(1)n

Find E(Xn)andVar(Xn)

Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

E(Xn)=E[E(XnXn1)]

By the given definition of conditional expectation

=E(Xn1)

=E[… (194 more words) …

Question number: 25

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of i=1nXi2 . What is its probability distribution?

Explanation

Let X ~ N (0, σ 2). The density function is

f(x)=12πσex22σ2

First we find the distribution of Y = X 2

F(y)=P[Yy]=… (243 more words) …

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