2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 −
x), 0 < x < 1,
0 elsewhere.
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Find P(−0.5 < X < 0.3).
(c) Find the median of X.
Let X and Y have the joint PDF
f(x) = { 1/2 0 < x + y < 2, x > 0, y > 0 ;
{ 0 elsewhere
a) sketch the support of X and Y
b) Are X and Y independent? Explain.
c) Find P(x<1 and y<1.5)
Let X and Y have the joint probability density function (pdf):
f(x, y) = 3/2 x2(1 − y), − 1 < x < 1, − 1 < y < 1
Find P(0 < Y < X).
Find the respective marginal pdfs of X and Y. Are X and Y independent?
Find the conditional pdf of X give Y = y, and E(X|Y = 0.5).
1) The PDF of a Gaussian random variable is given by fx(x).
fx(x)= (1/(3*sqrt(2pi)
)*e^((x-4)^2)/18
determine
a.) P(X > 4) b). P(X > 0). c). P(X
< -2).
2) The joint PDF of random variables
X and Y is given by
fxy(x,y)=Ke^-(x+y), x>0 , y>0
Determine
a. The constant k.
b. The marginal PDF fX(x).
c. The marginal PDF fY(y).
d. The conditional PDF fX|Y(x|y). Note
fX|Y(x|y) =
fxy(x,y)/fY(y)
e. Are X
and Y independent.
1) Let X be a continuous random variable. What is true about
fX(x)fX(x)?
fX(2) is a probability.
fX(2) is a set.
It can only take values between 0 and 1 as input.
fX(2) is a number.
2) Let X be a continuous random variable. What is true about
FX(x)FX(x)?
FX(x) is a strictly increasing function.
It decreases to zero as x→∞x→∞.
FX(2) is a probability.
FX(x) can be any real number.
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0.
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 Crameer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
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 or -(x/θ)^2 if you cannot read that)
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).
Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. (i)
Find the conditional means of X given Y, and Y given X. (ii) Find
the conditional variance of X given Y. (iii) Find the correlation
coefficient between X and Y.