Let X1, X2, X3, X4, X5 be independent continuous random
variables having a common cdf F and pdf f, and set p=P(X1 <X2
<X3 < X4 < X5).
(i) Show that p does not depend on F. Hint: Write I as a
five-dimensional integral and make the change of variables ui =
F(xi), i = 1,··· ,5.
(ii) Evaluate p.
(iii) Give an intuitive explanation for your answer to (ii).
Let X1 and X2 have the joint pdf
f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere
(a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1
= x1.
(b) Find the conditional expectation and variance of X1|X2 = x2 and
X2|X1 = x1.
(c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and
P(0 < X1 < 1/2).
(d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that
var(Y ) ≤ var(X2).
let X1, X2, X3 be random variables that are defined as
X1 = θ + ε1
X2 = 2θ + ε2
X3 = 3θ + ε3
ε1, ε2, ε3 are independent and the mean and variance are the
following random variable
E(ε1) = E(ε2) = E(ε3) = 0
Var(ε1) = 4
Var(ε2) = 6
Var(ε3) = 8
What is the Best Linear Unbiased Estimator(BLUE) when estimating
parameter θ from the three samples X1, X2, X3
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]
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1
+ X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y =
(Y1,Y2,Y3)′ using : Multivariate normal distribution
properties.
Let X and Y be two jointly continuous random variables with
joint PDF
f(x,y) = Mxy^2
0<x<y<1
a) Find M = ?
b) Find the marginal probability densities.
c) P( y> 1/2 | x = .25) = ?
d) Corr (x,y) = ?
Let X1, X2, X3, . . . be independently random variables such
that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is
independent of X1, X2, . . .. Further define T = XN
.
(a) Find E(T) and argue that T is short proper.
(b) Find the pgf of T.
(c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0.
(d) Use the pgf of...
(i) Find the marginal probability distributions for the random
variables X1 and X2 with joint pdf
f(x1, x2) =
12x1x2(1-x2) , 0 <
x1 <1 0 < x2 < 1
, otherwise
(ii) Calculate E(X1) and
E(X2)
(iii) Are the variables X1
and X2 stochastically independent?
(iv) Given the variables in the
question, find the conditional p.d.f. of X1 given
0<x2< ½ and the conditional expectation
E[X1|0<x2< ½ ].
2.2.8. Suppose X1 and X2 have the joint pdf
f(x1, x2) = "
e−x1 e−x2
x1 > 0, x2
> 0
0 elsewhere
.
For constants w1 > 0 and w2 > 0, let W = w1X1 + w2X2.
(a) Show that the pdf of W is
fW (w) = "
1
w1−
w2
(e−w/w1 − e−w/w2) w > 0
0 elsewhere
.
(b) Verify that fW (w) > 0 for w > 0.
(c) Note that the pdf fW...
Let X1,…, Xn be a sample of iid random variables with pdf f (x;
?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ.
Determine
a) L(?1, ?2).
b) the MLE of ?⃗ = (?1, ?2).
c) E(? ̂ 2).