Expextation-of-A-Rand-Variabl

Expectation of a Discrete Random Variable

A random variable X is a function from a sample S to real numbers R, i.e X: S——>R. Suppose you are tossing three coins then the sample space would be.

S={HHH,TTH, THT, THH, HTT, HTH, HHT, HHH}

Then X maps sample space to real numbers.

Where  you can see that R={0,1,2, 3}

You can also write probability distributions

f(0)= P(X=0)= 1/8

f(1)= P(X=1)= 3/8

f(2) = P(X=2)=3/8

f(3)= P(X=3)=1/8

P(X=0) denotes probability of  0 HEAD

P(X=1) denotes probability of one HEAD

P(X=2) denotes probability of two HEADs

P(X=3) denotes probability of three HEADs

And function f is probability density function, a probability density function has two properties.

(1)- f(x)≥0

(2) Σx f(x)=1 = 1/8+3/8+3/8+1/8

Expectation of a discrete random variable is defined as composition of random variable values weighted by the corresponding probabilities.

i.e.

E(X)= 0. (1/8 )+ 1. (3/8)+ 2. (3/8) + 3.(1/8)

E(X)=12/8=3/2=1.5

 

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