Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 14 - 19 of 72

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Question number: 14

» Probability » Probability Generating Functions

Appeared in Year: 2009

Essay Question▾

Describe in Detail

Given the distribution function

F(x)=12+x2(1+ x )<x<

find its probability density function.

Explanation

The distribution function written in this form

F(x)={12+x2(1+x);0<x<12x2(1+x);<x<0

F(x)… (98 more words) …

Question number: 15

» Probability » Convergence » In Distribution

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Show that convergence in probability implies convergence in distribution.

Explanation

The sequence X n converges to X in probability if for any ε > 0

P(XnXε)0asn

The sequence X n converges to the distribution of X as n tends to infinity if

Fn(… (303 more words) …

Question number: 16

» Probability » Probability of M Events Out of N

Appeared in Year: 2009

Essay Question▾

Describe in Detail

A die is rolled twice. Let A, B, C denote the events respectively that the sum of scores is 6, the sum of scores is 7, and the first score is 4. Are A and C independent? Are B and C independent?

Explanation

A 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), (3,… (229 more words) …

Question number: 17

» Probability » Tchebycheffs Inequality

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X be a random variable with E [X] = 4 and E [X 2] = 20. Use Chebyshev’s inequality to determine a lower bound for the probability P [0 < x < 8].

Explanation

Let X be a random variable with mean µ and variance σ 2. Then any k > 0, the Chebyshev’s inequality is

P(Xμkσ)11k2

or P(Xμ<k)>1… (75 more words) …

Question number: 18

» Probability » Probability Generating Functions

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

n=0snP(Xn)=P(s)1s

Explanation

To show this we use the left hand side, express the summation term

n=0snP(Xn)=P(X0)+sP(X1)+s2P(X2)… (352 more words) …

Question number: 19

» Probability » Bayes' Theorem

Appeared in Year: 2014

Essay Question▾

Describe in Detail

There are three identical bags U 1, U 2 and U 3. U 1 contains 3 red and 4 black balls; U 2 contains 4 red and 5 black balls; U 3 contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red. Find the probability that the first bag is chosen.

Explanation

Given that there are three identical bags U 1, U 2 and U 3. Then probabilities of selecting a bag are,

P(U1)=13;P(U2)=13;P(U3)=13

Let… (173 more words) …

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