
Probabilities assigned to various outcomes in S
in turn determine probabilities associated with
the values of any particular random variable X.

The probability distribution of X gives how
the total probability of 1 is distributed among
the various X values.
2

To give the probability distribution or
probability mass function (pmf) of a
discrete rv X, we give the values that X
can be, and the probability of taking on
those values.
3

Consider rolling a pair of die. Then S contains
the set of 36 ordered pairs {(1,1), (1,2), …,
(6,6)}

Let X=sum of values on the die, e.g. X(4,3)=7.

What is the probability distribution of X?
4

The Cal Poly Department of Statistics has a
lab with six computers reserved for
statistics majors. Let X denote the number
of these computers that are in use at a
particular time of day.

The table on the next slide gives the
probability of each value.
5

x
0
1
2
3
4
5
6
p(x)
.05
.10
.15
.25
.20
.15
.10
We can use elementary properties of
probability to calculate other probabilities of
interest. As examples, P( X  2)  .05  .10  .15  .30
P(2  X  5)  .15  .25  .20  .15  .75
6



Six lots of components are ready to be
shipped by a certain supplier. The number of
defective components in each lot is as follows:
Lot
1
2
3
4
5
6
#
defective
0
2
0
1
2
0
One of these lots will be picked at random to
be shipped.
Then p(0)=3/6=1/2; p(1)=1/6; p(2)=2/6=1/3
7

The pmf of a Bernoulli rv X is of the form:
1  p

P  x, p    p
0


if x  0
if x  1
otherwise
Each value of p yields a different pmf. p is
called a parameter of the distribution.
8

Suppose p(x) depends on a quantity that can
be assigned any one of several possible
values, with each value giving a different
probability distribution. The quantity is called
a parameter of the distribution.

The collection of probability distributions for
different values of the parameter is called a
family of probability distributions.
9

We observe the gender of each newborn child at a
hospital until a boy is born. Let p=P(B), and
assume that successive births are independent.
Then
1  p  x1 p x  1,2,3,
p( x)  
otherwise
0
p  0.5 may be appropriate for this situation, but
p  0.85
may be more appropriate if we are
looking for the first child with RH-positive blood.
This is the family of geometric distributions.
10

The cumulative distribution function (cdf)
F(x) of a discrete rv with pmf p(x) is defined
for every number x by
F ( x)  P ( X  x)   y:y x p ( y )
11

For
x
0
1
2
3
4
5
6
p(x)
.05
.10
.15
.25
.20
.15
.10
x0
0,
.05, 0  x  1

.15, 1  x  2

.30, 2  x  3
F ( x)  
.55, 3  x  4
.75, 4  x  5

.90, 5  x  6
1,
6 x
12

The probability mass function is given by the
size of the jumps of the cumulative
distribution function.
13

For any two numbers a and b with a  b ,
P  a  X  b   F b   F  a 
a
where
represents the largest possible X
value that is strictly less than a . If only integer
values are possible for a and b, then
P  a  X  b   F  b   F  a  1
14