Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with...
Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]
Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10...
Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...
Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and...
Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?
Let X1, X2,... be a sequence of
independent random variables distributed exponentially with mean 1.
Suppose...
Let X1, X2,... be a sequence of
independent random variables distributed exponentially with mean 1.
Suppose that N is a random variable, independent of the Xi-s, that
has a Poisson distribution with mean λ > 0. What is the expected
value of X1 + X2 +···+
XN2?
(A) N2
(B) λ + λ2
(C) λ2
(D) 1/λ2
5. Let X1, X2, . . . be independent random variables all with
mean E(Xi) =...
5. Let X1, X2, . . . be independent random variables all with
mean E(Xi) = 7 and variance
Var(Xi) = 9. Set
Yn =
X1 + X2 + · · · + Xn
n
(n = 1, 2, 3, . . .)
(a) Find E(Y2) and E(Y5).
(b) Find Cov(Y2, Y5).
(c) Find E (Y2 | X1).
(d) How should your answers from parts (a)–(c) be modified if the
numbers “2”, “5”, “7” and
“9” are replaced by m,...
Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let...
Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let
S = X1 + X2 + X3 + X4. Suppose we know that S is a Chi-Square
random variable with 2 degrees of freedom. What
is the distribution of each of the Xi?
Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the...
Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?
f X1,X2,X3,X,X5 are independent and identically distributed
geometrically distributed ran
dom variables with the parameter p,...
f X1,X2,X3,X,X5 are independent and identically distributed
geometrically distributed ran
dom variables with the parameter p, compute (a) Find c.d.f. of
Ymax = max{X1,...,X5};
(b) Find p.d.f. of Ymin = min{X1,...,X5}