Question

Let X ∼ N (0; σ2) and Y = | X |, where σ> 0. Obtain the density function probability of the Y random variable

Answer #1

Let X ∼ N(μ,σ2). Let Y = aX for some constant a. Find the joint
moment generating function of (X, Y ).

Consider the joint density function f (x, y) = 1 if 0<=
x<= 1; 0<=y<= 1. [0 elsewhere]
a) Obtain the probability density function of the v.a Z, where Z =
X^2.
b) Obtain the probability density function of v.a W, where W =
X*Y^2.
c) Obtain the joint density function of Z and W, that is, g (Z,
W)

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0,
otherwise } . a. Let W = max(X, Y ) Compute the probability density
function of W. b. Let U = min(X, Y ) Compute the probability
density function of U. c. Compute the probability density function
of X + Y .

Let X be a random variable with probability density function fX
(x) = I (0, 1) (x). Determine the probability density function of Y
= 3X + 1 and the density function of probability of Z = - log
(X).

Suppose that X1,..Xn are iid
N(0,σ2) where σ>0 is the unknown parameter. with
preassigned α in (0,1), derive a level α LR test for the null
hypothesis H0: σ2 = σ02
against H1: σ2/=σ02 in
the implementable form. (Hint: When is the function g(u) =
ue1-u, u > 0 increasing? When is it decreasing? Is
your test one-sided or two-sided?)

Let X be a random variable with probability density function
f(x) = {3/10x(3-x) if 0<=x<=2
.........{0 otherwise
a) Find the standard deviation of X to four decimal
places.
b) Find the mean of X to four decimal places.
c) Let y=x2 find the probability density function
fy of Y.

Let X be a gamma random variable with parameters α > 0 and β
> 0. Find the probability density function of the random
variable Y = 3X − 1 with its support.

Let X be the mean of a random sample of size n from a N(θ, σ2)
distribution,
−∞ < θ < ∞, σ2 > 0. Assume that σ2 is known. Show that
X
2 − σ2
n is an
unbiased estimator of θ2 and find its efficiency.

Let random variable X ∼ U(0, 1). Let Y = a + bX, where a and b
are constants.
(a) Find the distribution of Y .
(b) Find the mean and variance of Y .
(c) Find a and b so that Y ∼ U(−1, 1).
(d) Explain how to find a function (transformation), r(), so
that W = r(X) has an exponential distribution with pdf f(w) = e^
−w, w > 0.

1. Let fX(x;μ,σ2) denote the probability density function of a
normally distributed variable X with mean μ and variance σ2.
a. What value of x maximizes this function?
b. What is the maximum value of fX(x;μ,σ2)?

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