Show that ?̅ is an unbiased estimator for ?. ( please show all work completely)

Solutions

Expert Solution

We have to show that E ( $\bar{x}$ ) = $\mu$

Now,

$\bar{x}$ = $\sum$ Xi / n , i = 1 , 2 , 3 , ...

So,

E( $\bar{x}$ ) = E( $\sum$ Xi / n )

= 1 / n * E( $\sum$ Xi )

= 1 / n * [ E( X1 + X2 + X3 + ... + Xn) ]

= 1/n * [ E(X1) + E(X2) + E(X3) + ... + E(Xn) ]

Since X1 , X2 , X3 , ... Xn are random variables and expected value for each is equal which is $\mu$ ,

= 1/n * ( $\mu$ + $\mu$ + $\mu$ + ....n times)

= 1 / n ( n $\mu$ )

= $\mu$

Therefore, $\bar{x}$ is an unbiased estimator of $\mu$ .

orchestra answered 1 year ago

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