Expectation in Statistics

Expectation in Statistics

Expectation  in statistics

Expectation  in statistics is the weighted average of a random variable with its probability.

Suppose you toss three coins, then think of event to turn heads up.

Random variable associates number of occurrence of event to its probability.
i.e.
Expectation of random variable X is denoted by E(X).

1- Expectation of a Discrete Random Variable X

To understand expectation of a discrete random variable. Take an example of tossing of three coins.

The possible values when you toss three coins.
TTT
TTH
THT
THH
HTT
HTH
HHT
HHH

X(TTT)=0
X(TTH)=1
X(THT)=1
X(THH)=2
X(HTT)=1
X(HTH)=2
X(HHT)=2
X(HHH)=3

Then
P(X=0)=1/8
P(X=1)=3/8
P(X=2)=3/8
P(X=3)=1/8
Here, P(X) is probability distribution function of random variable X.

Then expression for expectation of a discrete random variable is.

Expectation  is the weighted average of a random variable with its probability.  Suppose you toss three coins, then think of  event to turn heads up.  Random variable associates number of occurence of event to its probability. i.e.  Expectation of random variable X is denoted by E(X).  <h2>1- Expectation of  a Discrete Random Variable X</h2>  The possible values  when you toss three coins. TTT TTH THT THH HTT HTH HHT HHH   X(TTT)=0 X(TTH)=1 X(THT)=1 X(THH)=2 X(HTT)=1 X(HTH)=2 X(HHT)=2 X(HHH)=3  Then  P(X=0)=1/8 P(X=1)=3/8 P(X=2)=3/8 P(X=3)=1/8 Here, P(X) is probability distribution function of random variable X.  Then expected value of random variable X is.  Image-1    <h2>1- Expectation of  a Continuous  Random Variable X</h2>  If X is a continuous random variable and  p(x) is a continuous probability distribution function. Then expectation of continuous random variable  X is.   Let p(x) be defined as   p(x)=3/2 x(x-1) 0<=x<=2 for a random variable X  Then   <h2> Properties of Expectation of a Random Variable</h2> If X and Y are independent random variables 1- E(X+Y)= E(X) + E(Y)  2- E(X.Y)= E(X) E(Y) 3- Is a is a and b are constants then 1- E(aX+b)= a E(X) + b  <h2> Variance of a Ranom variable X </h2> Variance of a random variable is  <h2>Covariance between Two Random Variables X and Y</h2> Covariance between Two Random Variables X and Y is

Then expected value of random variable X is.

Expectation in Statistics Discrete Random Variable

1- Expectation of a Continuous Random Variable X

If X is a continuous random variable and p(x) is a continuous probability distribution function.

Then expectation of continuous random variable X is.

Expectation of a Continuous Random Variable Formula

Let p(x) be defined as

p(x)=3/2 x(x-1) 0<=x<=2
for a random variable X.

Then expectation of continuous random variable X is

Expectation of a Continuous Random Variable Formula

Properties of Expectation of a Random Variable

If X and Y are independent random variables
1- E(X+Y)= E(X) + E(Y)
2- E(X.Y)= E(X) E(Y)
3- Is a is a and b are constants then
1- E(aX+b)= a E(X) + b

Variance of a Random variable X

Variance of a random variable is X

Variance of a Random Variable X

 

Covariance between Two Random Variables X and Y

Covariance between Two Random Variables X and Y is

Variance of a Random Variable X

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