Let ? and ? be two independent uniform random
variables such that
?∼????(0,1) and
?∼????(0,1).
A) Using the convolution formula, find the pdf
??(?) of the random variable
?=?+?, and graph it.
B) What is the moment generating function of ??
1) Let U1, U2, ... be independent random variables, each
uniformly distributed over the interval (0, 1]. These random
variables represent successive bigs on an asset that you are trying
to sell, and that you must sell by time = t, when the asset becomes
worthless. As a strategy, you adopt a secret number \Theta and you
will accept the first offer that's greater than \Theta . The offers
arrive according to a Poisson process with rate \lambda = 1....
Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1)
random variables. (
a) Compute the cdf of Y := min(X1, . . . , Xn).
(b) Use (a) to compute the pdf of Y .
(c) Find E(Y ).
Suppose A and B are independent uniform random variables on
[0,1]. What is the joint PDF of (R,S)? Prove that R and S are are
independent standard normal random variables.
R = sqrt(-2logA)*cos(2piB)
S = sqrt(-2logA)*sin(2piB)
Suppose A and B are independent uniform random variables on
[0,1]. What is the joint PDF of (R,S)? Prove that R and S are are
independent standard normal random variables.
R = sqrt(-2logA)*cos(2piB)
S = sqrt(-2logA)*sin(2piB)
You have three independent uniform random variables Xi on [0,1]
for i=1,2,3. calculate
(a) P(all of them are less than 1/2)
(b) P(at least one of them is less than 1/2)
(c) the conditional probability P(all of them are less than 1/2
| at least one of them is less than 1/2)
(d) the mean and the variance of S = X1 + X2 + X3
(e) P(the value of X2 lies between the values of the other two
random...
Let B = {u1,u2} where
u1 = 1 and u2 = 0
0 1
and
B' ={ v1 v2] where
v1= 2 v2= -3
1 4
be bases for R2
find
1.the transition matrix from B′ to B
2. the transition matrix from B to B′
3.[z]B if z = (3, −5)
4.[z]B′ by using a transition matrix
5. [z]B′ directly, that is, do not use a transition matrix
Let X,,X, and X, be independent uniform random
variables on [0,1] Write Y = X, +X, and Z = X+ X. a.) Compute
E[X,X,X,. (5 points) b.) Compute Var(X). (5 points) c.) Compute and
draw a graph of the density function fr. (15 points)