Let X1 and X2 be independent UNIF(0,1) random variables and
consider the transformations Y1= X1X2 and...
Let X1 and X2 be independent UNIF(0,1) random variables and
consider the transformations Y1= X1X2 and Y2 =X1/X2. Find the joint
pdf of Y1 and Y2 and indicate their joint support of Y1 and Y2.
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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.
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 ).
Let X1,X2,… be a sequence of independent random variables,
uniformly distributed on [0,1]. Define Nn to be the smallest k such
that X1+X2+⋯+Xn exceeds cn=n2+12n−−√, namely,
Nn
=
min{k≥1:X1+X2+⋯+Xk>ck}
Does the limit
limn→∞P(Nn>n)
exist? If yes, enter its numerical value. If not, enter
−999.
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6. Let X1, X2, ..., X101 be 101 independent U[0,1] random
variables (meaning uniformly distributed on the unit interval). Let
M be the middle value among the 101 numbers, with 50 values less
than M and 50 values greater than M.
(a). Find the approximate value of P( M < 0.45 ).
(b). Find the approximate value of P( | M- 0.5 | < 0.001 ),
the probability that M is within 0.001 of 1/2.
Consider independent random variables X1, X2, and X3 such that
X1 is a random variable having mean 1 and variance 1, X2 is a
random variable having mean 2 and variance 4, and X3 is a random
variable having mean 3 and variance 9.
(a) Give the value of the variance of
X1 + (1/2)X2 + (1/3)X3
(b) Give the value of the correlation of Y = X1- X2 and Z = X2 +
X3.
Let X1, X2, . . . be a sequence of independent and identically
distributed random variables where the distribution is given by the
so-called zero-truncated Poisson distribution with probability mass
function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2,
3...
Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is
independent of the Xi ’s.
1) Show that Y = X1
+X2 + ... + XN has a Poisson distribution
with mean nλ.
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 ??
2. Let X1, X2, . . . , Xn be independent, uniformly distributed
random variables on the interval [0, θ]. (a) Find the pdf of X(j) ,
the j th order statistic. (b) Use the result from (a) to find
E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the
mean difference between two successive order statistics. (d)
Suppose that n = 10, and X1, . . . , X10 represents the waiting
times that the n = 10...
Let X1 and X2 be independent standard
normal variables X1 ∼ N(0, 1) and X2 ∼ N(0,
1).
1) Let Y1 = X12 +
X12 and Y2 =
X12− X22 . Find the
joint p.d.f. of Y1 and Y2, and the marginal
p.d.f. of Y1. Are Y1 and Y2
independent?
2) Let W =
√X1X2/(X12
+X22) . Find the p.d.f. of W.
Let X1 and X2 be two independent random
variables having a chi-squared distribution with degrees of freedom
n1 and n2, respectively. Let
Y1 = (X1) / (X1 + X2)
and Y2 = X1 + X2
(a) Find the joint p.d.f. of Y1 and Y2
(b) Find the marginal p.d.f. of each of Y1 and
Y2
(c) Are Y1 and Y2 independent ?
Justify your answer.