Expected value

Expected value

Expected value or mean of an independent random discrete variable x with its mass function f(x):

Expected value or mean of an independent random continuous variable x with its density function f(x):

1. Mean of a function of random variables

Proof E(X+Y)=E(X)+E(Y) if X, Y are random variables

Regarding discrete variable

Regarding continuous variable:

Proof E(X*Y)=E(X)E(Y) if X and Y are iid.

Regarding discrete variable

Regarding continuous variable:

Proof E(aX+b)=aμ+b if E(X)=μ.

Regarding discrete variable

Regarding continuous variable:

Proof E(E(X))=μ if E(X)=μ.

2. Mean of sample mean

Proof

Similarly,

Proof E(S2)=σ2.

Unbiased estimator of σ2 is S2 if

Set E(X)=μ, Var(X)=E(x2)-E(x)2= σ2. So E(x2)=σ2+μ2. So

3. Mean of proportion

x is random variable. n is total number of trials.

The proportion of success is p=x/n