Let X,,X, and X, be independent uniform random
variables on [0,1] Write Y = X, +X, and Z = X+ X. a.) Compute
E[X,X,X,. (5 points) b.) Compute Var(X). (5 points) c.) Compute and
draw a graph of the density function fr. (15 points)
1. Let X and Y be independent random variables
with μX= 5, σX= 4,
μY= 2, and σY= 3.
Find the mean and variance of X + Y.
Find the mean and variance of X – Y.
2. Porcelain figurines are sold for $10 if flawless,
and for $3 if there are minor cosmetic flaws. Of the figurines made
by a certain company, 75% are flawless and 25% have minor cosmetic
flaws. In a sample of 100 figurines that are...
Let X and Y be independent Poisson random variables with
parameters 1 and 2, respectively, compute
P(X=1 and Y=2)
P(X+Y>=2)
Find Poisson approximations to the probabilities of the
following events in 500 independent trails with probabilities 0.02
of success on each trial.
1 success
2 or fewer success.
Let ?1, ?2, ?3 be 3 independent random variables with uniform
distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2,
?3}. Find the variance of ?2, and the covariance between the median
?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).
Let X and Y be two independent random variables. X is a binomial
(25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y.
a) Find the expected value of X, the expected value of Y, the
variance of X and the variance of Y.
b) Find the expected value of W.
c) Find the variance of W.
d) Find the covariance of Z and W.
d) Find the covariance of Z and W.
Let ?1 and ?2 be two independent random variables with uniform
distribution on [0, 1].
1. Write down the joint cumulative distribution function and
joint probability density function of ?1 + ?2 and ?1?2
Let ?1 and ?2 be two independent random variables with uniform
distribution on [0, 1].
1. Write down the joint cumulative distribution function and joint
probability
density function of ?1 + ?2 and ?1?2.
2. Write down the covariance between ?1 + ?2 and ?1?2.
3. Let ? be the largest magnitude (absolute value) of a root of the
equation
?^2 − ?1? + ?2 = 0. Let ? be the random event that says that
the
equation ?^2 −?1?...