Question

The random variable X is distributed with pdf f_{X}(x,
θ) = c*x*exp(-(x/θ)^{2}), where x>0 and θ>0.

Calculate the probability of X_{1} < X_{2},
i.e. P(X_{1} < X_{2}, θ).

Answer #1

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 )
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

The random variable X is distributed with pdf fX(x,
θ) = (2/θ^2)*x*exp(-(x/θ)2), where x>0 and
θ>0. Please note the term within the exponential is
-(x/θ)^2 and the first term includes a θ^2.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?

The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 )
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).

3. Let X be a continuous random variable with PDF
fX(x) = c / x^1/2, 0 < x < 1.
(a) Find the value of c such that fX(x) is indeed a PDF. Is this
PDF bounded?
(b) Determine and sketch the graph of the CDF of X.
(c) Compute each of the following:
(i) P(X > 0.5).
(ii) P(X = 0).
(ii) The median of X.
(ii) The mean of X.

Let X be a random variable with pdf given by fX(x) = Cx2(1−x)1(0
< x < 1), where C > 0 and 1(·) is the indicator
function.
(a) Find the value of the constant C such that fX is a valid
pdf.
(b) Find P(1/2 ≤ X < 1).
(c) Find P(X ≤ 1/2).
(d) Find P(X = 1/2).
(e) Find P(1 ≤ X ≤ 2).
(f) Find EX.

Consider a random sample X1,
X2, ⋯ Xn from the
pdf
fx;θ=.51+θx, -1≤x≤1;0,
o.w., where (this distribution arises in particle
physics).
Find the method of moment estimator of θ.
Compute the variance of your estimator. Hint: Compute the
variance of X and then apply the formula for X, etc.

5. Let X be a continuous random variable with PDF
fX(x)= c(2+x), −2 < x < −1,
c(2−x), 1<x<2,
0, elsewhere
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Determine the CDF of X and sketch its graph.
(c) Find P(X < 1.5).
(d) Find m = π0.5 of X. Is it unique?

A random variable X has the pdf given by fx(x) = cx^-3, x .>
2 with a constant c. Find
a) the value of c
b) the probability P(3<X<5)
c) the mean E(X)

Let X be a continuous random variable with a PDF of the form
fX(x)={c(1−x),0,if x∈[0,1],otherwise.
Find the following values.
1. c=
2. P(X=1/2)=
3. P(X∈{1/k:k integer, k≥2})=
4. P(X≤1/2)=

Let X ∼ Geo(?) with Θ = [0,1].
a) Show that pdf of the random variable X is in the
one-parameter
regular exponential family of distributions.
b) If X1, ... , Xn is a sample of iid Geo(?) random variables
with
Θ = (0, 1), determine a complete minimal sufficient statistic
for ?.

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