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A sample of size $n$ is drawn from $\{1,2, \cdots , N \}$ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate

<p>$(a)$ the  PMF of $X,$</p>

<p>$(b)$ $\mathrm E (X),$</p>

<p>$(c)$ $\mathrm {Var} (X),$</p>

<p>$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.</p>

I have found

$$P(X=k)= \frac {(N-k+1)^n - (N-k)^n} {N^n}.$$

But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.

Thank you very much.

I have found the joint probability of the random variables $X$ and $Y$ as follows $:$

$$P(X=i,Y=j) = \frac {(j-i+1)^n - 2(j-i)^n+(j-i-1)^n} {N^n}.$$ for $i=1,2,\cdots,N$ and $j=1,2,,\cdots,N$.

Is it correct? Please verify it.

little o
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1 Answers1

1

Your answer for part (a) is correct.

For part (b), write your result from part (a) as

$$\mathsf P(X\gt k)=\frac{(N-k)^n}{N^n}$$

and calculate

$$ \mathsf E(X)=\sum_{k=0}^{N-1}\mathsf P(X\gt k)=\sum_{k=1}^N\frac{k^n}{N^n}\;, $$

which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)\approx\frac N{n+1}$.

For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.

Your answer for part (d) is correct; it was recently asked about in
Calculating probability of High+Low = T on N dice with S sides.

joriki
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