Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 63 - 67 of 72

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Question number: 63

» Probability » Moment Generating Functions

Appeared in Year: 2014

Essay Question▾

Describe in Detail

X 1, X 2, …, X N are independently, identically distributed random variables. Define S N = X 1 + X 2 + … + X N , where N is a random variable independent of X i, i = 1, 2, … N.

Show that the moment generating function (mgt) of S N is

MSN(t)=MN[logMX(t)]

where My (t) is the mgf of a random variable Y. Hence find the mgf of S N when N follows a Poisson distribution with parameter λ. and X i follows an exponential distribution with mean parameter θ, i = 1 to N.

Explanation

The moment generating function of S N is where S N =X 1 +X 2 +…+X N, X i are i. i. d random variable and N is also a random variable.

MSN(t)=E(etSN)

=E(… (209 more words) …

Question number: 64

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2011

Essay Question▾

Describe in Detail

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: 65

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

Describe in Detail

The joint density of (X, Y) is

f(x,y)={x2+xy3;0<x<1,0<y<20,otherwise

Find the conditional densities and E [X|Y = 1.5].

Explanation

First find the marginal distribution of X and Y

f(x)=02f(x,y)dy=02(x2+xy3)dy

=[x2y+xy26]… (215 more words) …

Question number: 66

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

Describe in Detail

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: 67

» Probability » Probability Generating Functions

Appeared in Year: 2009

Essay Question▾

Describe in Detail

Find the generating function of X whose probability density function is

P [X = r] =pq r-1, r = 1, 2, …, 0 < p < 1, q = 1 - p

Explanation

The probability generating function is defined as

E(tz)=z=1tzP[X=z]

=z=1tzpqz1

=ptz=1[tq… (28 more words) …

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