Two fair dice are rolled. Let X be the product of the number of
dots that...
Two fair dice are rolled. Let X be the product of the number of
dots that show up.
(a) Compute P(X = n) for all possible values of n.
(b) Compute E(X).
(c) Compute Var(X) and SD(X).
Two fair dice are rolled at once. Let x denote the difference in
the number of dots that appear on the top faces of the two dice.
For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so
x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the
probability distribution for x. Arrange x in increasing order and
write the probabilities P(x) as simplified fractions.
Suppose two fair dice are rolled. Let X denote the product of
the values on the dice and Y denote minimum of the two dice.
Find E[X] and E[Y]
Find Var X and Var Y
Let Z=XY. Find E[Z].
Find Cov(X,Y) and Corr(X,Y)
Find E[X|Y=1] and E[Y|X=1]
Two fair dice are rolled and the outcomes are recorded. Let X
denotes the larger of the two numbers obtained and Y the smaller of
the two numbers obtained. Determine probability mass functions for
X and Y, and the cumulative distribution functions for X and for Y.
Present the two cumulative distribution functions in a plot.
Calculate E (2X + 2Y −8).
2. Three fair dice are rolled. Let X be the sum of the 3
dice.
(a) What is the range of values that X can have?
(b) Find the probabilities of the values occuring in part (a);
that is, P(X = k) for each k in part (a). (Make a table.)
3. Let X denote the difference between the number of heads and
the number of tails obtained when a coin is tossed n times.
(a) What are the possible...
two fair dice are each rolled once. Let X denote the absolute
value of the difference between the two numbers that appear.
List all possible values of X
Find the probability distribution of X.
Find the probabilities P(2<X<5) and P(2£X<5).
Find the expected value mand standard deviation of X.
Two fair dice are rolled:
a) What is the probability of an even number or a 3 on the first
die? Are these two events mutually exclusive and why?
b) What is the probability of an even number on the first die
and a 5 on the second? Is conditional probability involved in this
case? Why or why not?
Three fair dice are rolled. Let S be the total number of spots
showing, that is the sum of the results of the three rolls.
a) Find the probabilities P(S = 3), P(S = 4), P(S = 17), P(S =
18).
b) Find the probability P(S ≥ 11).
Two dice are rolled. Let the random variable X denote
the number that falls uppermost on the first die and let Y
denote the number that falls uppermost on the second die.
(a) Find the probability distributions of X and
Y.
x
1
2
3
4
5
6
P(X = x)
y
1
2
3
4
5
6
P(Y = y)
(b) Find the probability distribution of X +
Y.
x + y
2
3
4
5
6
7
P(X...
Two fair dice are thrown. Let X be the number of 5’s
and Y be the number of 6’s.
Find (a) the joint PMF of X and Y,
(b) the two marginal distributions (PMFs),
and (c) the conditional distribution (PMF) of Y given
X = x for each possible value of X.