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From a box containing N identical tickets numbered 1 to N, n tickets are drawn without replacement. Let X be the largest number drawn. Find E[X]

I got the pdf as x-1Cn-1/NCn And E(X)=summation x*pdf Here i am stuck and i dont know whether what i have done is right or no

user292986
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  • What have you tried and where did you get stuck? Add this to your question by means of an edit. – drhab Nov 24 '15 at 08:52
  • Can you find $P(X\leq k)$ for $k\in{n,n+1,\dots,N}$? If so then you can apply my hint. – drhab Nov 24 '15 at 09:18
  • Your probability mass function (pmf, not pdf) is not okay. The best way to find it is: $P(X=k)=P(X\leq k)-P(X\leq k-1)$. Actually if you use my hint you don't have to find it. Finding $P(X\leq k)$ is enough then. – drhab Nov 24 '15 at 09:32

1 Answers1

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Hints:

  • $\mathbb EX=P(X\geq1)+\dots+P(X\geq N)$
  • $P(X\geq k+1)+P(X\leq k)=1$

edit:

$$\mathbb{E}X=\sum_{i=1}^{N}ip_{i}=\sum_{i=1}^{N}\sum_{k=1}^{i}p_{i}=\sum_{k=1}^{N}\sum_{i=k}^{N}p_{i}=\sum_{k=1}^{N}P\left(X\geq k\right)=$$$$\sum_{k=1}^{N}\left(1-P\left(X\leq k-1\right)\right)=N-\sum_{k=1}^{N}P\left(X\leq k-1\right)=N-\sum_{k=n}^{N-1}P\left(X\leq k\right)$$ and for $k\in\{n,n+1,\dots,N\}$: $$P(X\leq k)=\frac{\binom{k}{n}}{\binom{N}{n}}$$

drhab
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