Question

Q3. The aim of this question is to see a typical use-case for the linearity of expectation. Consider an experiment in which we toss a biased coin (probability of heads = p) n times. Let Y be the random variable that is the number of heads. Also, let X; be the 0/1 random variable that is 1 if the ith toss was heads and otherwise.

A good explanation of how to answer (b) and (c) will be upvoted :)

Answer 1

$X_i=1$ if head in ith toss and 0 otherwise.

It is given probability of heads is p.

So we get $P(X_i=1)=P(\text{head in ith toss})=p$ .so that

$P(X_i=0)=P(\text{tail in ith toss})=1-P((\text{head in ith toss}))=1-p$.

So the expected value,

$E(X_i)=1P(X_i=1)+0P(X_i=0)\\ =1\times p+0\times (1-p)=p$

Now, again $X_i=1$ if head in ith toss and 0 otherwise.

So $\sum_{i=1}^n X_i$ is the total number of heads as $X_i=1$ for heads and $X_i=0$ for non-heads. That means we add one only if head occurs.

Also it is given as total number of heads in n tosses. So we can write

$Y=\sum_{i=1}^n X_i$

So

$E(Y)=E(\sum_{i=1}^n X_i)\\ =\sum_{i=1}^n E(X_i) \text{ using linearity of expectation}\\ =\sum_{i=1}^n p \quad , \text{ by part b}\\ =p \sum_{i=1}^n 1=p\timesn=np$