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STATISTICS 122: Probability Theory II
Review Exercises
1. True or False.
a. (x,y) is a mass point of the random vector (X,Y)’ if and only if P(X=x, Y=y) >0.
b. Multinomial distribution : k+1 :: Generalized Hypergeometric distribution : k+2.
c. If FX,Y(x,y) is continuous at the point (1,2), then pX,Y(x,y) = 0.
d. If (X,Y)’ is a jointly discrete random vector, then X and Y are discrete random variables.
e. Knowledge of the joint CDF implies knowledge of the marginal PMF or PDF.
2. Three balanced coins are tossed independently. One of the variables of interest is Y1, the number of
heads. Let Y2 denote the amount of money won on a side bet in the following manner. If the first head
occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3,
respectively. If no heads appear, you lose $1.
a. Show that (Y1,Y2)’ is a random vector.
b. Find the joint probability function for Y1 and Y2.
c. Obtain the joint distribution of Y1 and Y2 using your answer in (b).
d. What is the probability that fewer than three heads will occur and you will win $1 or less?
3. If the joint probability distribution of X and Y is given by: pX ,Y ( x, y ) 
a. Find P(X > 2, Y  1).
b. Find (X > Y).
c. P(X + Y = 4)
d. P(X  2, Y=1)
xy
I{0,1,2,3} ( x )I{0,1,2} ( y )
30
e. Are X and Y independent?
4. From a sack of fruits containing 3 oranges, 2 apples and 2 bananas, a random sample of 4 pieces of
fruit is selected. If X is the number of oranges and Y is the number of apples in the sample,
a. Find the joint probability distribution of X and Y.
b. What is the probability that the sum of the number of oranges and banana is less than 2?
5. An electronic system has one each of two different types of components in joint operation. Let X and Y
denote the random lengths of life (in hours) of the components of type I and type II, respectively. The
joint density function is given by: f X ,Y ( x, y ) 
1 ( x2 y )
xe
I(0, ) ( x )I(0, ) ( y ) .
8
a. Find the distribution of the length of life of the Type I component.
b. What is the probability that the electronic system is still working after an hour?
6.
kxy ,0  x  1,0  y  1
Let X and Y have the joint probability density function given by: f X ,Y ( x, y )  0, otherwise

a. Find the value of k that makes this a valid probability density function.
b. Obtain the joint distribution function for X and Y.
c. Evaluate P(X ≤ 1/2, Y ≤ 3/4).
7. Suppose that X and Y are uniformly distributed over the unit square. Calculate P(X < -ln(Y)).
30 xy 2 , x  1  y  1  x,0  x  1
f
(
x
,
y
)


8. The joint density function of X and Y is given by: X ,Y
.
0, elsewhere
a.
b.
c.
d.
e.
Show that the marginal density of X follows a Beta distribution with a = 2, b = 4.
Derive the marginal density of Y.
Find the conditional density of Y given X = x.
Evaluate the P(Y > 0 | X = ¾).
Are X and Y independent? Why or why not?

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