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Chapter 4: Q.4.13 (page 170)

Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, ... , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P{X = k}. This is known as the method of maximum likelihood estimation.

Short Answer

Expert verified

In the given information the answer isp=kn

Step by step solution

01

Step 1:Given Information

We are require to findp∈0,1which maximize the probability

P(X=k)=nkpk·(1-p)n-k

where kis some fixed number. consider the logarithm of expression we have

logP(X=k)=lognk+k·logp+(n-k)log(1-p)

02

Step 2:Calculation

We need to maximize the function p→k.

logp+(n-k)log(1-p).call the functiong.

we have that

dgdp=kp-n-k1-p=0

since, equation is equal tok(1-p)=(n-k)p.which is satisfied forp=kn.

03

Step 3:Final Answer

The answer isp=kn.

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