Question

Moments and Function Generator of Moments

An urn contains 4 red balls, 3 blue, 2 green and one yellow.
Three balls are obtained from this sample (without replacement).
Let X be the random variable that represents the number of red
balls that are extracted.

a) Find the probability function of the random variable X

b) Find the first moment of the random variable

c) Find the second moment of the random variable

d) Find the third moment of the random variable

e) Find the fourth moment of the random variable

f) Find the moment generator function.

Answer #1

An urn contains 10 balls, 2 red, 5 blue, and 3 green balls. Take
out 3 balls at a random,
without replacement. You win $2 for each green ball you select and
lose $3 for each red ball you
select. Let the random variable X denote the amount you win,
determine the probability mass
function of X.

An urn contains 1 white, 2 black, 3 red, and 4 green balls. If 6
balls are selected randomly (without replacement) and X represents
the number of selections that are either red or green, find: (a)
the probability mass function for X. (b) the expected value of X
(calculate this value directly by using the probability mass
function from part a).

2. Urn A contains 6 green and 4 red balls, and Urn B contains 3
green and 7 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

We have three urns: the first urn has 6 red balls and 4 green
balls; the second urn has 15 red balls and 5 green balls and the
third urn has 20 red balls and 10 green balls. We pick 4 balls from
the first urn (sampling with replacement); we select 5 balls from
the second urn (sampling with replacement) and we select 10 balls
from the third urn (sampling with replacement). Let X1 denote the
number of red balls...

Urn A contains 5 green and 4 red balls, and Urn B contains 3
green and 6 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

Urn A contains 6 green and 4 red balls, and Urn B contains 3
green and 7 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

An urn has six balls, 3 are red, 2 are blue, and 1 is green. You
choose 2 at random, without replacement. Let X be
the number of red balls drawn, and Y be the number of green balls
drawn.
(A) Find the joint distribution of X and Y.
(B) Find the marginal distribution of X and Y.
(C) Find the conditional distribution of Y, given that X=1.
(D) E[X], E[Y], E[X2], E[Y2], E[XY], AND
Cov(X,Y).

Urn A contains 5 green and 3 red balls, and Urn B contains 2
green and 6 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

An urn contains five blue, six green and seven red balls. You
choose five balls at random from the urn, without replacement (so
you do not put a ball back in the urn after you pick it), what is
the probability that you chose at least one ball of each
color?(Hint: Consider the events: B, G, and R, denoting
respectively that there are no blue, no green and no red balls
chosen.)

An urn contains 4 red balls and 3 green balls. Two balls are
sampled randomly.
Let Z denote the number of green balls in the sample when the
draws are done without replacement. Give the possible value of Z
and its probability mass function (PMF).

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