Question

1. Let P = (X, Y ) be a uniformly distributed random point over the diamond with vertices (1, 0),(0, 1),(?1, 0),(0, ?1). Show that X and Y are not independent but E[XY ] = E[X]E[Y ]

Answer #1

X is uniformly distributed on the interval (0, 1), and Y is
uniformly distributed on the interval (0, 2). X and Y are
independent. U = XY and V = X/Y .
Find the joint and marginal densities for U and V .

Let X and Y be independent; each is uniformly distributed on [0,
1]. Let Z = X + Y. Find:
E[Z|X]. Your answer should be a function of x.

1 point) Let A, B, and C be independent random variables,
uniformly distributed over [0,5], [0,15], and [0,2] respectively.
What is the probability that both roots of the equation Ax2+Bx+C=0
are real?

Consider the triangle with vertices (0,0),(3,0),(0,3) and let P
be a point chosen uniformly at random inside the triangle. Let X be
the distance from P to (0,0). (i) Is X a random variable? Explain
why or why not. (ii) Compute P(X≥0), P(X≥1), P(X≥2), and
P(X≤3).

Suppose that X is a random variable uniformly distributed over
the interval (0, 2), and Y is a random variable uniformly
distributed over the interval (0, 3). Find the probability density
function for X + Y .

Let ?, ?, and ? be independent random variables, uniformly
distributed over [0,5], [0,1] , and [0,2] respectively. What is the
probability that both roots of the equation ??^2+??+?=0 ar
e real?
P
.S read careful, I had already waste a chance to post question
in this

Let X and Y be independent random variables, uniformly
distribued on the interval [0, 2]. Find E[e^(X+Y) ].

Let X1,...,X99 be independent random variables, each one
distributed uniformly on [0, 1]. Let Y denote the 50th largest
among the 99 numbers. Find the probability density function of
Y.

Let X and Y be random variables, P(X = −1) = P(X = 0) = P(X = 1)
= 1/3 and Y take the value 1 if X = 0 and 0 otherwise. Find the
covariance and check if random variables are independent.
How to check if they are independent since it does not mean that
if the covariance is zero then the variables must be
independent.

A two dimensional variable (X, Y) is uniformly distributed over
the square having the vertices (2, 0), (0, 2), (-2, 0), and (0,
-2).
(i) Find the joint probability density function.
(ii) Find associated marginal and conditional distributions.
(iii) Find E(X), E(Y), Var(X), Var(Y).
(iv) Find E(Y|X) and E(X|Y).
(v) Calculate coefficient of correlation between X and Y.

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