Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 32 - 37 of 72

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Question number: 32

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

Essay Question▾

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Let X be a positive valued random variable. Prove that

P[Xr]E(X)r,r>0

Hence deduce the Chebychev’s inequality.

Explanation

The expectation of X is define as

E(X)=xf(x)dx

For any x ≥ r

{xr}xf(x)dx

r{xr}… (63 more words) …

Question number: 33

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

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Let Y 1 denote the first order statistic in a random sample of size n from a distribution that has the pdf

f(x)={e(xθ),θ<x<0,otherwise

Obtain the distribution of Z n =n (Y 1 -θ).

Explanation

To find the distribution of order statistic, the pdf is

fx(j)(x)=n!(j1)!(nj)!f(x)[F(x)]j1[1F… (290 more words) …

Question number: 34

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2013

Essay Question▾

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Find the distribution of the ratio of two iid random variables with density function:

f(x)=ex,0<x<0,elsewhere

Explanation

Let consider x and y are two iid random variable with density function is same. Then find the distribution of their ratio that is x/y. The joint pdf of x and y is

f(x,y)=e(x+y)

To find this… (190 more words) …

Question number: 35

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2012

Essay Question▾

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12.3 % of the candidates in a public examination score at least 70%, while another 6.3 % score at most 30%. Assuming the underlying distribution to be normal, estimate the percentage of candidates scoring 80 % or more.

Explanation

Let total marks obtain is 100. Assuming the underlying distribution to be normal, the mean µ and variance σ 2. It is given that

P(X70)=P(z70μσ2)=0.123

The value of z corresponding to… (173 more words) …

Question number: 36

» Probability » Probability of M Events Out of N

Appeared in Year: 2013

Essay Question▾

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A fair die is rolled twice. Let A be the event that the first throw shows a number ≤ 2, and B be the event that the second throw shows at least 5. Show that P (AUB) =5/9.

Explanation

A fair die is rolled twice, the sample space consists of thirty six outcomes. The sample space is

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4),… (174 more words) …

Question number: 37

» Probability » Probability of M Events Out of N

Appeared in Year: 2014

Essay Question▾

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A positive integer X is selected at random from the first 50 natural numbers.

Calculate P (X + 96/X > 50).

Explanation

Total number of possible outcomes = number of ways in which the one random number can be chosen out of 50

n=(501)=50

Let Y be the event whose satisfied the relation, X + 96/X > 50

For which X we choose is

X… (93 more words) …

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