Probability-Expectation (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 6 of 6

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Question number: 1

» Probability » Expectation

Appeared in Year: 2014

Essay Question▾

Describe in Detail

In a lottery 1000 tickets are sold and the cost of a ticket is if 10. The lottery offers a first prize of if 1,000, two second prizes of if 500 each, and three third prizes of if 100 each. A person purchases a ticket. If X denotes the amount he may get, find E (X) and V (X).

Explanation

The probability of purchases a ticket is

First prize amount is 1000

Two second prizes amount is 500 each

Three third prizes amount is 100 each

X denotes the amount he may get

Then, the expected value of X is

The variance of X is

… (119 more words) …

Question number: 2

» Probability » Expectation

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let (X, Y) have a bivariate distribution with finite moments upto order 2. Show that

(i) E (E (X|Y) ) =E (X)

(ii) V (X) ≥ V (X|Y)

Explanation

Let (X, Y) have a bivariate distribution with density function id f (x, y). The conditional expectation is define as

Note that E (X|y) is a function of y. If we allow y to vary over the support of Y, then E (X|y) as a function of the random variable Y. Taking the expectation of this function over the random variable Y

… (79 more words) …

Question number: 3

» Probability » Expectation

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

Explanation

We known that in continuous random variable, the mean is

We know that F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

Hence it’s proofed.

… (158 more words) …

Question number: 4

» Probability » Expectation

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Let X 1, X 2, ···, X n be random variables such that

Also

Find

Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

By the given definition of conditional expectation

.

.

.

The variance is

… (182 more words) …

Question number: 5

» Probability » Expectation

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that E (X - a) 2 is minimized for a = E (X), assuming that the· first 2 moments of X exist.

Explanation

Assume Y = E (X - a) 2

Here we want to minimized Y for a that is

The value of E (X - a) 2 is minimum when the value of a is E (X).

… (74 more words) …

Question number: 6

» Probability » Expectation

Appeared in Year: 2015

Essay Question▾

Describe in Detail

For random variables X, Y, show that

Explanation

We know that

… (162 more words) …

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