Probability (ISS (Statistical Services) 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

Equation

find its probability density function.

Explanation

The distribution function written in this form

Equation

Equation

The density function is Equation

Equation

where Equation

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

Equation

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

Equation

For ε > 0,

Equation

Equation

Hence

Equation ……… (1)

Note that

Equation

Equation …. (2)

From

… (21 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,

… (153 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

Equation

or Equation

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Equation

Using Chebyshev’s inequality

… (2 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

Equation

Explanation

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

Equation

Equation

Equation

Equation

Equation

Equation

We know that the probability generating function is defined as

Equation

Putting this in the above equation, we prove the following

Equation

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,

Equation

Let X be the event of selecting a red ball

Probability of selecting a red ball in U 1 is

Equation

Similarly, Probability of selecting a

… (40 more words) …

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