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

Equation

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

Equation

Equation

The variance of X is

Equation

Equation

… (2 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

Equation

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

… (66 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:

Equation

Explanation

We known that in continuous random variable, the mean is

Equation

Equation

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

dF (x) =-dS (x)

Equation

Equation

Equation

Equation

Hence it’s proofed.

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

Equation

Also

Equation

Find Equation

Explanation

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

Equation

By the given definition of conditional expectation

Equation

Equation

Equation

.

.

.

Equation

Equation

The variance is

Equation

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

Equation

Equation

Equation

Equation

Equation

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

Question number: 6

» Probability » Expectation

Appeared in Year: 2015

Essay Question▾

Describe in Detail

For random variables X, Y, show that

Equation

Explanation

We know that

Equation

Equation

Equation

Equation

Equation

Equation

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