Probability-Expectation (ISS 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

P=11000

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

E(X)=1000… (90 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

E(Xy)=xfXY(xy)dx

Note that E (X|y) is a function of y. If… (340 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:

E(X)=0(1F(x))dx0F(x)dx

Explanation

We known that in continuous random variable, the mean is

E(X)=xf(x)dx

=0xf(x)dx+0xf(x)dx

We… (135 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

E(X1)=μ,E(Xr Xr1)=Xr1;r=2(1)n

Also

E(X1μ)2=σ2,E[(XrXr1)2 Xr1]=σ2;r=2(1)n

Find E(Xn)andVar(Xn)

Explanation

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

E(Xn)=E[E(XnXn1)]

By the given definition of conditional expectation

=E(Xn1)

=E[… (194 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

ya=0

aE(Xa)2=0

aE(X22aX+a2… (60 more words) …

Question number: 6

» Probability » Expectation

Appeared in Year: 2015

Essay Question▾

Describe in Detail

For random variables X, Y, show that

V[Y]=EX[V(Y X)]+VX[E(Y X)]

Explanation

We know that

V[Y]=E(Y2)[E(Y)]2

=EX[E(Y2)X][E(Y)]2

=EX[V(YX)+… (104 more words) …

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