Properties of expectation and variance

1. Properties of expectation and variance

1.1 Property one

When there is only one random variable $X$, Use the definition of expectation to calculate $E[a_1{g}_1(X)+a_2{g}_2(X)]$

The definition of expectation


When there are two different random variables $X=X_1+X_2$

1.2 Property two

If $g_i(X_i)=c$, that $E[g_i(X_i)]=c$

1.3 Property three

First give the definition of variance

1.4 Property 4

Calculation $E[2{\mu }_{X}X]$ Temporal associative property $E[aX]=aE[X]$, so $E[2{\mu }_{X}X]=2{\mu }_{X}E[X]$
Calculation $E[{\mu }_X^2]$ Temporal associative property $E[c]=c$, so $E[{\mu }_X^2]={\mu }_X^2$

1.5 Nature five

Calculation $E[{\mu }_X]$ Temporal associative property $E[c]=c$, so $E[{\mu }_X]={\mu }_X$

1.6 Property VI

Prove the above formula according to the definition of variance

1.7 Property 7