Question

Help please all parts

Answer 1

a) p = x/n

b) p = 2/5

c) see attached

Step-by-step explanation:

a) You apparently want to maximize L(p) with respect to p, assuming values of x and n are fixed. The derivative of L(p) can be found using the product rule. Then the value of p that maximizes L(p) can be found by setting the derivative to zero.

`(dL)/(dt)=(d)/(dt)\left(p^(x)(1-p)^((n-x))\right)=xp^((x-1))(1-p)^((n-x))-(n-x)p^(x)(1-p)^((n-x-1))\\\\=L(p)(x-np)/(p(1-p))`

Setting this to zero, we have ...

`x-np=0\\\\x=np\\\\p=(x)/(n)`

b) For x=2, n=5, the likelihood function L(p) is maximized for ...

... p = x/n = 2/5

c) See the attached plot. The peak (maximum likelihood) is at p=0.4 = 2/5.

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Comment on the problem

This problem involves derivatives, something that you don't expect in a middle school math curriculum.