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The given problems are from Mathematical Statistics. Kindly solve the problems below with detail explanation. Please...

The given problems are from Mathematical Statistics. Kindly solve the problems below with detail explanation. Please do not use short cut terms. Neat and clean writing and explanation will be helpful. The problems are:

(1) It is not known what proportion p of the purchases of a certain brand of breakfast cereal are made by women and what proportion are made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and 12 were made by men. Find the M.L.E of p.

(2) Suppose that X1, X2, ........Xn form a random sample from a Poisson distribution for which the mean m is unknown, (m > 0).

a. Determine the M.L.E of m, assuming that at least one of the observed values is different from 0.

(3) Suppose that X1, X2, ........Xn form a random sample from a normal distribution for which the mean M is known, but the variance (sigma)2 is unknown. Find the M.L.E of

(sigma)2 .

(4) Suppose that X1, X2, ........Xn form a random sample from a distribution for which the pdf f(x|0) is as follows:

f(x|0) = { mxm - 1 for 0 < x < 1,

{ 0 Otherwise

Also, suppose that the value of m is unknown (m > 0). Find the M.L.E of m.

Math   Statistics-And-Probability   
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