A box contains $N$ tickets numbered $1, 2, 3,…., N$. If $m$ tickets are drawn one by one from the box without replacement, then find the mean of the sum of the numbers obtained on the tickets drawn.
I have approached the sum as below.
Let $X_i$ denote the number on the $i$th ticket drawn, where $i= 1, 2,…, m$.
The sum of the numbers obtained on the tickets drawn is $S= \sum_{i=1}^{m}X_i$
Hence, the required mean
$= E(S)
=E(\sum_{i=1}^{m}X_i)
=\sum_{i=1}^{m}E(X_i).$
Each $X_i$ can take the values $1, 2,…, N$ with probability $\frac{1}{N}$.
Then $E(X_i)= \frac{N+1}{2}$
So, $E(S)= \frac{m(N+1)}{2}$
But my doubt is in the above line 'Each $X_i$ can take the values $1, 2,…, N$ with probability $\frac{1}{N}$.'
Because when the drawing is done without replacement, after each draw, the number of tickets remaining decreases by 1. So the number of values left for $X_2$ is $N-1$, not $N$.
So how can the probability be $\frac{1}{N}$?
Will anyone please explain where is the mistake?
Thanks in advance.
Best Answer
The probability is indeed not $\frac1N$ if you have drawn at least one ticket. You have to look at the probability to draw a combination. It is a good idea is to look on a small example.
Let´s say a box contains $3$ tickets numbered $1,2,3$. The probabilities to draw number $i$ at the first draw are $P(X_1=1)=P(X_1=2)=P(X_1=3)=\frac13$.
Thus $\mathbb E(X_1)=\sum\limits_{i=1}^3 p(x_i)\cdot x_i=\frac13\cdot 1+\frac13\cdot 2+\frac13\cdot 3=2$
Now what is $\mathbb E(X_2)$?
The probability to draw the combinations $(i,j)$ are
$P(X_2=(1,2))=\frac13\cdot \frac12\cdot 2=\frac13$, $P(X_2=(1,3))=\frac13\cdot \frac12\cdot 2=\frac13$, $P(X_2=(2,3))=\frac13\cdot \frac12\cdot 2=\frac13$
The corresponding means are $1.5,2,2.5$. It follows that the expected value is
$\mathbb E(X_2)=\frac13\cdot (1.5+2+2.5)=2$
It is much more difficult to make a proof for an arbitrary $N$. But you can use the Linearity of expectation.
For a uniform, discrete random variable we have $\mathbb E(X_1)=\frac{N+1}2$. Thus $\mathbb E(X_i)=\frac{N+1}2 \ \ \forall \ 1\leq i\leq N$