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
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
Question number: 2
» Probability » Expectation
Appeared in Year: 2011
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… (66 more words) …
Question number: 3
» Probability » Expectation
Appeared in Year: 2013
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.
Question number: 4
» Probability » Expectation
Appeared in Year: 2012
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
Question number: 5
» Probability » Expectation
Appeared in Year: 2011
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).
Question number: 6
» Probability » Expectation
Appeared in Year: 2015
Describe in Detail
For random variables X, Y, show that
Explanation
We know that