STAB52 F23
Quiz 4
Name (f,l):
ID #:
1. (5 points) Consider the cumulative distribution function (CDF)
(
2
1 − e−x /2 x > 0
F (x) =
,
0
x≤0
Find the corresponding probability density function (PDF) and verify that it integrates
to 1.
2. Consider the joint probability mass function (PMF)
(
c(x + y), ∀ x = 1, 2, . . . , 10, y = 1, 2, . . . , 10
pX,Y (x, y) =
0,
otherwise
(a) (5 points) Find the value of the constant c so that this is a valid PMF.
(b) (5 points) Find the marginal PMF of Y .
(c) (5 points) Find the probability that X is less than or equal to Y , i.e., find
P (X ≤ Y ) = P ((X, Y ) ∈ {(x, y) : 1 ≤ x ≤ y ≤ 10})
(Hint:
Pn
x=1
x2 = n3 /3 + n2 /2 + n/6.)