ish are caught in lake until fish of a certain type been point. n tained. Let X
ID: 2966166 • Letter: I
Question
ish are caught in lake until fish of a certain type been point. n tained. Let X enote the total number of fish caught at that does Assume that the lake is so large that the extraction of these fish not change the proportion p of this type of fish in the lake. Show that the random variable X possesses the density function given by p"(1 p)x x n, n 1, where n is assumed to be a 1) is an 1)/(X fixed integer. Use this result to show that biased estimator of p. Note that this estimator differs from the intuitive estimator n/x, which is biased for this problemExplanation / Answer
Suppose we have caught x fishes.
The last fish being caught is the certain type, and the x-1 others contains n-1 certain type.
So we have C(x-1, n-1) possible combination here to get n-1 certain type out of the x-1 remaining fishes.
Since we have caught n fishes of a certain type with probability p
and x-n fishes of the other type with probability 1-p, we get :
P(X=x) = C(x-1,n-1) * p^n * (1-p)^(x-n)
Now by definition of the expectation :
E((n-1)/(X-1))
= sum(x >= n) (n-1)/(x-1) * C(x-1 ,n-1) p^n (1-p)^(x-n)
= sum(x >= n) C(x-2, n-2) p^n (1-p)^(x-n) (since k/n C(n,k) = C(n-1,k-1))
= p^n ( 1 +(1-n)(p-1) + n(n-1)/2! (p-1)^2 - (n-1)n(n+1)/3! (p-1)^3 + ...)
Let a = 1-n and x=p-1 then
= p^n * ( 1+ax+a(a-1)/2! x^2 + a(a-1)(a-2)/3! x^3 + ...)
= p^n (1+x)^a (binomial taylor expansion )
= p^n ( 1 + p-1)^(1-n)
= p^n * p^(1-n)
= p
Hence proved.
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