Question

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| . Calculate E(Y1) and E(Y2), and compare them.

Answer 1

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)=\frac1{√(2\pi)}e^(-x^2/2)`

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

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

`\implies E[Y_2]=n√(\frac2\pi)`