Solve the following:
Q1) A coin, having probability t p of landing heads, is flippe duntil head apears for the rth time. Let N denote the number of flips required.
a) Calculate E[X] for the maximum random variable for Exercise 37.
b) Calculate E[X] for X as in Exercise 33.
c) Calculate E[X] for X as in Exercise 34.
Q2)In deciding upon the appropriate premium to charge, insurance companies sometimes use the exponential principle, defined as follows.
With X as the random amount that it will have to pay in claims, the premium charged by the insurance company is
P=1/a In(E[e^ax])
where a is some specified positive constant. Find P when X is an exponential random variable with parameter (lambda), and a=a(lambda),
where 0<(alpha)<1
Q3) Let X be a random variable with probability density
f(x)={c(1-x^2), -1
0 otherwise
(a) What is the value of c?
(b) What is the cumulative distribution function of X?
Q4) Let the probability density of X be given by
f(x)={c(4x-2x^2), 0
otherwise
0(this zero is right under the c and is enclosed in the bracket)
(a) What is the value of c?
(b) P{1/2
Q5) Let X1, X2, ...., Xn be independent random variables, each having a uniform distribution function of M, Fm (.), is given by
Fm(x)=x^n, 0(less than or equal to)x(less than or equal to) 1
What is the probability density function of M?