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.
Let X be a continuous random variable with pdf: f(x) = ax^2 −
2ax, 0 ≤ x ≤ 2
(a) What should a be in order for this to be a legitimate
p.d.f?
(b) What is the distribution function (c.d.f.) for X?
(c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)?
(d) What is the 90th percentile value of this distribution?
(Note: If you do this problem correctly, you will end up with a
cubic...
Let X have the pdf fX(x) = 3(1 − x) 2 , 0 < x < 1.
(a) Find the pdf of Y = (1 − X) 3 . Specify the distribution of
Y (name and parameter values). (b) Find E(Y ) and Var(Y ).
Let X be a continuous random variable with a probability density
function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) =
e −x
a. Find the expression for the probability density function fY
(y).
b. Find the domain of the probability density function fY
(y).
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).
(15 pts) Suppose that the continuous random variable X has
pdf
?(?) = {
?; 0 < ? < 2 2?; 5 < ? < 10 0; otherwise
a) Determine the value of c that makes this a legitimate pdf.
b) Sketch a graph of this pdf.
c) Determine the cumulative distribution function (cdf) of X.
d) Sketch a graph of this cdf.
e) Calculate ? = ?(?) and ? = ??(?).
f) What is ?(? = ?)?
g) Compute...
Let X and Y be continuous random variables with joint pdf
f(x, y) = kxy^2 0 < x, 0 < y, x
+ y < 2
and 0 otherwise
1) Find P[X ≥ 1|Y ≤ 1.5]
2) Find P[X ≥ 0.5|Y ≤ 1]