2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a...
(b) Proof of if and only if (IFF) Va > 0,6> 0 € N, the greatest common divisor of a and b is b if and only if bla.
11) Find the value of k so that f (x) = kx4(1 -x)2 for 0 <x < 1 is a beta density function Also determine find E(T) and Var (T)
1. Solve the following IBVPs for a string of unit length, subject to the given conditions 100, t) = 0, u(1,t)-0, t>0 a) b) a(x, 0) = f(x), ut (x,0) = g(x) 0 < x < 1 f(x) = sinocoso, g(x) = 0, c = 1/л. f(x) = sinx + (1/2)sin3m + 3sin7m, g(x) = snLTX, C = 1 0, if 0sxs 30( 3 30
A continuous random variable X has a beta distribution with
p.d.f :
1 f(x) = 0<<<1, a > 2 B(4, 5)22-1(1 – 2)8-1 Determine E (3) HINT: E possible. (-) + E(X) Please show your work and simplify your final answer as much as
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).
6. For the probability density function given by +1) -1<x<1, compute, using the definition the mean and variance of the distribution.
7. Let X, X, be a random sample with common pár 1 2 f(x) θ e-A, x > 0,0 > 0, 0 elsewhere. (a) Find the maximum likelihood estimator of θ, denoted by (b) Determine the sampling distribution of θ (c) Find Eô) and Var(). (d) What is the maximum value of the likelihood function? θ .
Problem 2 If the cumulative distribution function of X is given by o F(b) = b<0 0<b<1 1<b<2 2<b<3 3<b<3.5 b> 3.5 1 calculate the probability mass function of X.
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
1. Given a continuous random number x, with the probability density P(x) = A exp(-2x) for all x > 0, find the value of A and the probability that x > 1.