Discrete Random Variables: Probability Distribution Function (PDF) for a Discrete Random Variable
1.6
2008/05/16 14:38:08 GMT-5
2008/07/31 10:40:58.319 GMT-5
Barbara
Illowsky
illowskybarbara@deanza.edu
Susan
Dean
deansusan@deanza.edu
Connexions
cnx@cnx.org
discrete
distribution
elementary
function
PDF
probability
random
statistics
variable
This module introduces the Probability Distribution Function (PDF) and its characteristics.
A discrete probability distribution function has two characteristics:
- Each probability is between 0 and 1, inclusive.
- The sum of the probabilities is 1.
A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times a newborn wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5.
P(X) = probability that X takes on a value x.
X
P(x)
0
P(X=0)=250
1
P(X=1)=1150
2
P(X=2)=2350
3
P(X=3)=950
4
P(X=4)=450
5
P(X=5)=150
X takes on the values 0, 1, 2, 3, 4, 5.
This is a discrete PDF because
- Each P(X) is between 0 and 1, inclusive.
- The sum of the probabilities is 1, that is,
2
50
+
11
50
+
23
50
+
9
50
+
4
50
+
1
50
=
1
Suppose Nancy has classes 3 days a week. She attends classes 3 days a week 80% of the time, 2 days 15% of the time, 1 day 4% of the time, and no days 1% of the time.
Let X = the number of days Nancy ____________________ .
Let X = the number of days Nancy attends class per week.
X takes on what values?
0, 1, 2, and 3
Construct a probability distribution table (called a PDF table) like the one in the previous example. The table should have two columns labeled X and P(X). What does the P(X) column sum to?
X
P(x)
0
0.01
1
0.04
2
0.15
3
0.80
Probability Distribution Function (PDF)
A mathematical description of a discrete random variable (RV), given either in the form of the equation (by formula) , or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome.
A biased coin with probability 0.7 of head is tossed 5 times. We are interested in the number of heads (means, the RV X = the number of heads). X is Binomial RV, so X ∼B
5
.
7
and
P
(
X
=
x
)
=
5
x
.
7
x
.
3
5
−
x
or in the form of the table.
x
P
(
X
=
x
)
0
0.0024
1
0.0284
2
0.1323
3
0.3087
4
0.3602
5
0.1681