(i) Find the marginal probability distributions for the random
variables X1 and X2 with joint pdf
f(x1, x2) =
12x1x2(1-x2) , 0 <
x1 <1 0 < x2 < 1
, otherwise
(ii) Calculate E(X1) and
E(X2)
(iii) Are the variables X1
and X2 stochastically independent?
(iv) Given the variables in the
question, find the conditional p.d.f. of X1 given
0<x2< ½ and the conditional expectation
E[X1|0<x2< ½ ].
The random variables ? and ? have the following joint pdf. ??,?
(?, ?) = ?? -8x^2-18y^2
a) Find the mean and variance of ? and ? and the value of ?.
b) Determine if ? and ? are independent.
c) Determine the distribution of ? and ?.
(1 point) Let ?1 and ?2 have Poisson distributions with the same
average rate λ = 0.6 on independent time intervals of length 1 and
3 respectively. Find Prob(?1+?2=3) to at least 6 decimal
places.
Let X and Y be continuous random variables with joint pdf
f(x, y) = kxy^2 0 < x, 0 < y, x
+ y < 2
and 0 otherwise
1) Find P[X ≥ 1|Y ≤ 1.5]
2) Find P[X ≥ 0.5|Y ≤ 1]
Using binomial and poisson distributions to find probabilities
using the two equations from binominal and poisson
An inspector looks for blemishes in the finish on porcelain and
counts the number of defects on batches of 12 vases. If the mean is
1.2, what is the probability he finds exactly one blemish?
(poisson)
A nail salon tracks the number of customers that enter each
minute. In an average minutes 0.35 customers enter. What is the
probability at least 1 customer enters...
Let X1,...,Xn be exponentially distributed
independent random variables with parameter λ.
(a) Find the pdf of Yn=
max{X1,...,Xn}.
(b) Find E[Yn].
(c) Find the median of Yn.
(d) What is the mean for n= 1, n= 2, n= 3? What happens as n→∞?
Explain why.
Let X1 and X2 have the joint pdf
f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere
(a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1
= x1.
(b) Find the conditional expectation and variance of X1|X2 = x2 and
X2|X1 = x1.
(c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and
P(0 < X1 < 1/2).
(d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that
var(Y ) ≤ var(X2).
Let X and Y have joint PDF
f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y
< infinity
with parameters λ > 0 and μ > 0
a) Find c such that this is a PDF.
b) Show that X and Y are Independent
c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in
exponential form
d) Find the marginal distribution of Y, f(y)
e) Find E(Y)