Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 17 - 22 of 22

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Question number: 17

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

Essay Question▾

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Suppose that the random variable X has a normal distribution with mean µ and variance σ 2. Let φ be the distribution function of a standard normal variate. Find the density of φ (X-µ/σ). Also find E [φ (X-µ/σ) ].

Explanation

X has a normal distribution with mean µ and variance σ 2

f(x)=12πσe12(xμσ)2;<x<;<μ<

Let Z=X… (166 more words) …

Question number: 18

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

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Prove that for r = 1, 2, …, n

1Γrμtr1etdt=x=0r1eμμxx!

Explanation

We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.

P(Xr1)=x=0r1eμμxx!

P(X… (271 more words) …

Question number: 19

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2011

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Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:

Explanation

Let U = F (x), then the distribution function of G of U is given by

G(u)=P[Uu]=P[F(X)u]=P[XF1(u)]

Since… (284 more words) …

Question number: 20

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2011

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Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.

Explanation

Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is

MZ(t)=eγ(et1)

So, the sum of X and… (114 more words) …

Question number: 21

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

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Let the joint p. d. f. of (X, Y) be f (x, y) = e -y, 0 < x < y < ∞.

Obtain the probability P (X + Y ≤ 1).

Explanation

The Joint p. d. f. is

f (x, y) = e -y, 0 < x < y < ∞.

Let assume X + Y =U and Y = V, then X = U-V

Using Jacobian technique

J=xuxvy… (131 more words) …

Question number: 22

» Probability » Standard Probability Distributions » Exponential

Appeared in Year: 2009

Essay Question▾

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Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless 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… (218 more words) …

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