Question

You are given that X1 and X2 are two independent and identically distributed random variables with a Poisson distribution with mean 2. Let Y = max{X1, X2}. Find P(Y = 1).

Answer #1

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, . . . , 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]

Let
x1, x2 x3 ....be a sequence of independent and identically
distributed random variables, each having finite mean E[xi] and
variance Var（xi）.
a）calculate the var （x1+x2）
b）calculate the var（E[xi]）
c） if n-> infinite, what is Var（E[xi]）？

Continuous random variables X1 and X2 with joint density
fX,Y(x,y) are independent and identically distributed with expected
value μ.
Prove that E[X1+X2] = E[X1] +E[X2].

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, compute (a) Find c.d.f. of
Ymax = max{X1,...,X5};
(b) Find p.d.f. of Ymin = min{X1,...,X5}

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

Let X1 and X2 be independent random variables such that X1 ∼ P
oisson(λ1) and X2 ∼ P oisson(λ2). Find the distribution of Y = X1 +
X2.s

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=1) ?

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

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