Statistics : Random Variables and Mathematical Expectation

**Addition
and Multiplication Theorem on Expectations**

If
*X* and *Y* are two discrete random variables
then

(X+Y)
= *E*(*X*) + *E*(*Y*)

Let the random variable *X*
assumes the values *x*_{1}, *x*_{2} ... *x _{n}* with corresponding probabilities

Let *X* and *Y* are two continuous random variables
with probability density functions *f*(*x*) and* f*(*y*) respectively. Then* E *(*X
*+* Y*) =* E*(*X*) +* E*(*Y*)

**Proof**:

**1. Statement: **(*aX*+*b*) = *aE*(*X*)+*b* where *a* and *b* are constants.

**Proof: ***E*(*aX*+*b*)
= *E*(*aX*)+*E*(*b*) by property 3

=
*aE*(*X*)+*b* by property 2

(*aX*-*b*) = *aE*(*X*)-*b*

Find
the expectation of the sum of the number obtained on throwing two dice.

Let
*X*&*Y* denote the number obtained on the
I and II die respectively. Then each of them is a random variable which takes
the value 1,2,3,4,5 and 6 with equal probability 1/6.

Thus
the expectation of the numbers obtained on two dices.

*X*+*Y *takes the values 2, 3…12 with
their probability given by

Let
*X* and *Y* are two random variables with
p.d.f given by

*Solution:*

Statement:
If *X* and *Y* are two independent variables
then *E*(*XY*) = *E* (*X*) *E*(*Y)*

Statement:
If *X* and *Y* are independent random variables
Then *E* (*XY*) = *E*(*X*) *E*(*Y*)

Two
coins are tossed one by one. First throw is considered as *X*
and second throw is considered as *Y* following joint probability
distribution is given by,

[getting
Head is taken as 1 and Tail is taken as 0]

Verify
*E*(*XY*)= *E*(*X*) *E*(*Y*)

A
random variable *XY* can take the values 0 and 1

[It is applicable only when *X*
and *Y* are independent]

The independent random variables *X* and *Y* have the p.d.f
given by

Prove that
*E*(*XY*) =* E*(*X*)* E*(*Y*)

*X *and* Y *are independent

*f*(*x*,*y*) =* f*(*x*)
×* f*(*y*)

*f*(*x*,*y*) = 4*ax*×4*by*

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11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation : Addition and Multiplication Theorem on Expectations |

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