2 years ago

Help anyone?? I don’t get this

1 answer:

8 0

All you have to do is compare the magnitude or the number part of each value, in other words take the absolute value of each and compare the numbers only.

Absolute value of a negative number is positive.

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Let X1, . . . ,Xn be an i.i.d. random sample from a N(0, 1) population. Define Y1 = 1 n n X i=1 Xi , Y2 = 1 n n X i=1 |Xi| . Cal

nalin [4]

For each $1\le i\le n$ , $E[X_i]=0$ , so that

$\displaystyle E[Y_1]=E\left[\frac1n\sum_{i=1}^nX_i\right]=\frac1n\sum_{i=1}^nE[X_i]=0$

Meanwhile,

$\displaystyle E[Y_2]=\frac1n\sum_{i=1}^nE[|X_i|]$

Each of the $X_i$ have PDF

$f_{X_i}(x)=\dfrac1{\sqrt{2\pi}}e^{-x^2/2}$

for $x\in\Bbb R$ . From this we get

$E[|X_i|]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty|x|e^{-x^2/2}\,\mathrm dx=\sqrt{\frac2\pi}\int_0^\infty xe^{-x^2/2}\,\mathrm dx=\sqrt{\frac2\pi}$