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

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

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

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.

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

By the given definition of conditional expectation

.

.

.

The variance is

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).

We know that