Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 22

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Question number: 1

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

Essay Question▾

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If X ~ N (0, 1), obtain the distribution of X 2.

Explanation

X ~ N (0, 1). The density function is

f(x)=12πex22

Let assume Y = X 2

F(y)=P[Yy]=P[X2y]

=P[… (139 more words) …

Question number: 2

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2013

Essay Question▾

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Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.

Explanation

The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is… (315 more words) …

Question number: 3

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

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Prove that the sum of two independent chi-squared random variables is also chi-squared.

Explanation

Let X and Y are two independent chi-squared random variables with degree of freedom n and m respectively. We proof this by moment generating function. The moment generating function of chi-squared distribution is

MX(t)=(12t)n2

Then moment… (134 more words) …

Question number: 4

» Probability » Standard Probability Distributions » Negative Binomial

Appeared in Year: 2012

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Items from a large lot are examined one by one until r items with a rare manufacturing defect are found. The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).

Explanation

In this question, the sample size is n = x+r given and each trail only two possible outcomes. The probability of defect is same for each trail and trails are independent. The experiment continues until r defectives.

In the given question the number of manufacturing defect are fixed which is… (186 more words) …

Question number: 5

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2010

Essay Question▾

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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: 6

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2011

Essay Question▾

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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: 7

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

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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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