# Statistics basic question: Probability distrbution

## Homework Statement

A random variable X has three, and only three, possible values, 0; 2; 4
with the following probability distribution:

Probability:
Getting a 0 is 0:7

Getting a 2 is 0:1

Getting a 4 is 0:2

Let X1 and X2 be two independent random variables with this distribution.

(a) Determine the probability distribution for

X(bar)2 = (X1 + X2)/2
and use this distribution to determine the expectation and variance of X(bar)2

(b) Determine the expectation and variance of X and verify that the
variance of X(bar)2 is one-half the variance of X.

## The Attempt at a Solution

X2(bar) = 0 with probability 0.49
X2(bar) = 1 with probability 0.14
X2(bar) = 2 with probability 0.29
X2(bar) = 3 with probability 0.04
X2(bar) = 4 with probability 0.04

not sure how to work out the variance?

Regards as always
Tam

Last edited:

Ray Vickson
Homework Helper
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## Homework Statement

A random variable X has three, and only three, possible values, 0; 2; 4
with the following probability distribution:

Probability:
Getting a 0 is 0:7

Getting a 2 is 0:1

Getting a 4 is 0:2

Let X1 and X2 be two independent random variables with this distribution.

(a) Determine the probability distribution for

X(bar)2 = (X1 + X2)/2
and use this distribution to determine the expectation and variance of X(bar)2

(b) Determine the expectation and variance of X and verify that the
variance of X(bar)2 is one-half the variance of X.

## The Attempt at a Solution

X2(bar) = 0 with probability 0.49
X2(bar) = 1 with probability 0.14
X2(bar) = 2 with probability 0.29
X2(bar) = 3 with probability 0.04
X2(bar) = 4 with probability 0.04

not sure how to work out the variance?

Regards as always
Tam

You have the probability distribution of $\bar{X}_2.$ Just apply the _definition_ of variance and the formulas for it that you can find in any relevant book, or on-line if you look.

RGV