perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn? The Problem is: A perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn? 



I got to E of $S_n$ = $\sum_{n=1}^{+\infty} \space\space\space Sn (\frac{1}{2})^n$
Let's say a coin is tossed once: we get the Expectation of number of heads as 

E[R] = no.of heads*prob(no.of heads) = 1*1/2 = 1/2.

Now let n=2, E[R2] = 2*1/4 + 1*2/4 + 0*1/4 = 1. 

Now let n=3, E[R3] = 3*1/8 + 2*3/8 + 1*3/8 = 12/8 = 3/2.


As we keep continuing this we observe that E[Rn] = n*1/2. 

This is the linearity of Expectation. That is, E[R] = E[R1] + E[R2] + .. . + E[Rn].
Let's say a coin is tossed once: we get the Expectation of number of heads as 

E[R] = no.of heads*prob(no.of heads) = 1*1/2 = 1/2.

Now let n=2, E[R2] = 2*1/4 + 1*2/4 + 0*1/4 = 1. 

Now let n=3, E[R3] = 3*1/8 + 2*3/8 + 1*3/8 = 12/8 = 3/2.


As we keep continuing this we observe that E[Rn] = n*1/2. 

This is the linearity of Expectation. That is, E[R] = E[R1] + E[R2] + .. . + E[Rn].