Probability of a coin expresses in a "strange" way

Question

In a book , the probability of head in a coin it's expressed as:

$P=\frac{\left \langle N_{H} \right \rangle}{N}=0.5$

I thought that the probability of head was : $P=\frac{N_{H}}{N}$

What is the meaning of these "right angle parenthesis" ?

Answer 1

Assume that you have $N$ random variables $X_1,..,X_N$, where $X_i = 1$ if the coin is head and $0$ if it is tails. Then the number of heads out of $N$ coin tosses is $$ N_H = \sum_{i=1}^N X_i $$ and is a random quantity, since each $X_i$ is random. A different trial of $N$ coin tosses will give a different value for $N_H$.

The average number of heads is sometimes denotes by $\langle N_H \rangle$ is given by $$ \langle N_H \rangle = \mathbb{E} [N_H] = \sum_{i=1}^N \mathbb{E}[X_i] $$ If the probability of getting heads for each coin toss is $P$, then $$ \mathbb{E}[X_i] = P \cdot 1 + (1-P) \cdot 0 = P $$ and thus $$ \frac{\langle N_H \rangle}{N} = P $$ which is different from $$ \frac{N_H}{N} = \frac{1}{N} \sum_{i=1}^N X_i $$ which is a random.