mathematics, statistics

You observe k i.i.d. copies of the discrete uniform random variable Xi , which takes values 1 through n with equal probability.

Define the random variable M as the maximum of these random variables, M=maxi(Xi) .

1.) Find the probability that M≤m , as a function of m , for m∈{1,2,…,n} .

2.) Find the probability that M=1 .

3.) Find the probability that M=m for m∈{2,3,…n} .

4.) For n=2 , find E[M] and Var(M) as a function of k .

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  1. 5.) As k (the number of samples) becomes very large, what is E[M] in terms of n ?
    As k→∞ , E[M]→

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  2. Please help us!!!

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  3. 1. (m/n)^k
    2. (1/n)^k
    3. 1-(1/n)^k

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  4. 4. E[M] = 2−1/2^k, Var[M] = 1/2^k - 1/2^(2*k)
    5. E[M] = n

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  5. I think 1. is:
    P(M<=m)=(1/m)^k.

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  6. I think 1 is: (n-m+1)/n . There are always n numbers to choose from but only a subset of those will be >= the current max???

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  7. 3.) Find the probability that M=m for m∈{2,3,…n} .
    3. 1-(1/n)^k
    Shouldn't this probability depend on m somehow?
    P(M=m)=P(M<=m+1)-P(M<=m)=(m+1/n)^k-(m/n)^k
    ?

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  8. Yeah melo, your number 3 is clearly wrong. It is dependent on m, first of all

    Anon is right. P(M=m) = P(M<=m) - P(M<=m-1) = (m/n)^k - [(m-1)/n)]^k

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