(a) Given that X' is the mean of a random sample of size n , i.e. X1, X2, ..., Xn from a distribution with a mean miu and a variance sigma^2. Let W=(Xi - miu)/(sigma/sqrt(n)) .
(i) Show that
E(W)=(E(e^((t/sqrt(n))((Xi-miu)/sigma))).

Question

(a)	 Given that X'  is the mean of a random sample of size n , i.e. X1, X2, ..., Xn  from a distribution with a mean miu  and a variance sigma^2. Let W=(Xi - miu)/(sigma/sqrt(n)) .
		(i)  	Show that  
E(W)=(E(e^((t/sqrt(n))((Xi-miu)/sigma))).

holsteremission · Answer

Is $X'$ supposed to mean $\displaystyle\frac1n\sum_{i=1}^nX_i$ ?

And each of the $X_i$ are distributed with mean $\mu$ and variance $\sigma^2$, correct? (all equal for each $i$)

Also, are you sure you have the definition of $W$ written correctly?
$$W=\frac{X_i-\mu}{\dfrac\sigma{\sqrt n}}$$Did you mean $X'$ in place of $X_i$? Or is there additional notation that is missing?