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I am learning Variance.

$${\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[(X-\operatorname {E} [X])^{2}\right]\\[4pt]&=\operatorname {E} \left[X^{2}-2X\operatorname {E} [X]+\operatorname {E} [X]^{2}\right]\\[4pt]&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]\operatorname {E} [X]+\operatorname {E} [X]^{2}\\[4pt]&=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\end{aligned}}} $$

where, the part

$$\operatorname {E} \left[2X\operatorname {E} [X]\right] = 2\operatorname {E} [X]\operatorname {E} [X] $$

is a little bit difficult to justify, can anyone give a hint? which rule can apply this.

brennn
  • 155

2 Answers2

4

Linearity of Expected value applys this.

let $a = \operatorname{E}[X]$, which is a constant

your formula is now $\operatorname{E} \left[ 2X a \right] = 2\operatorname{E}[X] a = 2\operatorname {E} [X]\operatorname {E} [X]$

czlsws
  • 262
0

Hint: Note that $\mathrm{E}[X]$ is a constant, so can be pulled outside the expectation.

Minus One-Twelfth
  • 7,413