Question

Let A1, ... ,20 be independent events each with probability 1/2. Let X be the number of events among the first 10 which occur and let Y be the number of events among the last 10 which occur. Find the conditional probability that X = 5, given that X + Y = 12.

Answer #1

A fair coin is tossed three times. Let X be the number of heads
among the first two tosses and Y be the number of heads among the
last two tosses. What is the joint probability mass function of X
and Y? What are the marginal probability mass function of X and Y
i.e. p_X (x)and p_Y (y)? Find E(X) and E(Y). What is Cov(X,Y) What
is Corr (X,Y) Are X and Y independent? Explain. Find the
conditional probability mass...

Let A1, A2, . . . , An be n independent events in a sample space
Ω, with respective probability pi = P (Ai). Give a simple
expression for the probability P(A1 ∪A2 ∪...∪An) in terms of p1,
p2, ..., pn. Let us now apply your result in a practical setting: a
robot undergoes n independent tests, which are such that for each
test the probability of failure is p. What is the probability that
the robot fails at least...

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

Consider a succession of independent Bernoulli tests of
parameter p. Let X be the number of failure before the first
success and Y the number of failure between the first success and
the second.
a) Calculate the joint density function of X and Y.
b) Calculate the conditional density function of X given Y =
y.
c) Calculate the conditional density function of Y given X =
x.
d) Are the variables X and Y independent? Argue your
answer.

Let X be a number chosen at random from the set
{1, 2, ... ,20} and let Y be a number chosen at random
from the set {1, 2, ... , X }. Let
pX |Y (x|y)
denote the condition distribution of X, given that
Y = y. Find
pX |Y (19|18)

Let U and V be two independent standard normal random variables,
and let X = |U| and Y = |V|.
Let R = Y/X and D = Y-X.
(1) Find the joint density of (X,R) and that of (X,D).
(2) Find the conditional density of X given R and of X given
D.
(3) Find the expectation of X given R and of X given D.
(4) Find, in particular, the expectation of X given R = 1 and of...

A fair six-sided die is rolled 10 independent times. Let X be
the number of ones and Y the number of twos.
(a) (3 pts) What is the joint pmf of X and Y?
(b) (3 pts) Find the conditional pmf of X, given Y = y.
(c) (3 pts) Given that X = 3, how is Y distributed
conditionally?
(d) (3 pts) Determine E(Y |X = 3).
(e) (3 pts) Compute E(X2 − 4XY + Y2).

Let X and Y be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

Let X and Y be two continuous random variables with joint
probability density function f(x,y) = xe^−x(y+1), 0 , 0< x <
∞,0 < y < ∞ otherwise
(a) Are X and Y independent or not? Why?
(b) Find the conditional density function of Y given X = 1.(

2.
2. The joint probability density function of X and Y is given
by
f(x,y) = (6/7)(x² + xy/2),
0 < x < 1, 0 < y < 2. f(x,y) =0
otherwise
a) Compute the marginal densities of X and Y. b) Are X and Y
independent. c) Compute the conditional density
function f(y|x) and check restrictions on function you derived d)
probability P{X+Y<1} [5+5+5+5 = 20]

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