Hi! Please try to answer to these 5 questions in detail. Thank you.
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1) The random variable X has density function f(x|w) = 2x/w2 (w
square)..where 0<x<w
 and the prior density for w is f(w)= 1 ... o<w<1
 (a) How to find the posterior density for w assuming n=1. 
 (b) how to find Bayes estimator for w assuming the loss function
l(w*;w)=w2(w-w)2 {l(w- hat;w) = w square (w-w)square.} and n=1.

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2) A random variable X has p.d.f. (probability distribution function)
 f(x)=(k + 1)xk   {f(x) = (k + 1) x rest to k} where 0<x<1,
characterized by the parameter k. Assume that k has p.d.f. 1/(k + 1)2
{1/(k+1) square} on (o,infinity). Using quadratic loss function and
sample size of 2, with x1 (x one) = 1/2 and x2 (x two) = 2/3, test the
hypothesis
 H0 (Null): k <= 1.5
 H1: k > 1.5

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3) The dreaded foot-in-mouth syndrome (FIMS) strikes one in hundred
Engineering students. A new test for FIMS has 5 % error rate for false
negatives and 2 % for false positive. If you tested positively for
FMIS, what is the probability that you actually have it. (and how to
calculate that)?

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4) The island nation of hannibal has 1,000,000 inhabitants, of which
30% are adult males. A witness to a robbery notices that the adult
male thief had 6 fingers on his right hand. The island police know
that only 1 out of 100,000 males have this characteristic. An adult
male with 6 fingures on his right hand was picked up for questioning.
Based on the information alone, estimate the chances that the suspect
committed the crime.

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5) A balanced coin is tossed fairly until the first head appears. Your
winnings are based on the number of tosses that are made before the
game ends. If a head appears on the first toss, you wins $2. If no,
the pot is doubled to $4 - the reward if a head appears on a second
toss. The pot is doubled after every coin toss that results in a tail.
The winnings are $2 raised to the power of the number of tosses until
and including the first head.
 
(a) What are your expected winnings?
(b) How much would you be willing to pay for the privilege of playing this game.