Math 141 Lecture Notes
Section 8.1 Distributions of Random Variables
Random Variables
A random variable is a rule that assigns a number to each outcome of a chance
experiment.
It is a function with domain given by the set of outcomes of a chance experiment and
range contained in the set of real numbers.
Example 1: A coin is tossed four times. Let the random variable X denote the number
of heads that occur in the four tosses.
a. List the outcomes of the experiment; that is, find the domain of the function X.
b. Find the value assigned to each outcome of the experiment by the random
variable X.
c. Find the event comprising the outcomes to which a value of 2 has been assigned
by X. This event is written (X = 2) and is the event comprising the outcomes in
which two heads occur.
Example 2: A die is cast repeatedly until a four occurs. Let the random variable Y
denote the number of casts of the die in the experiment. What are the values of Y?
Example 3: A disposable flashlight is turned on until its battery runs out. Let the
random variable Z denote the length (in hours) of the life of the battery. What values may
Z assume?
A random variable is called finite discrete if it assumes only finitely many values.
A random variable is said to be infinite discrete if it takes on infinitely many values,
which may be arranged in a sequence.
A random variable is called continuous if the values it may assume comprise an interval
of real numbers.
Probability Distributions of Random Variables
The probability distribution of a random variable may be given in the form of a
formula or displayed in a table that gives the distinct (numerical) values of the random
variable and the probabilities associated with these values.
Example 4: Find the probability distribution of the random variable associated with the
experiment of Example 1.
Example 5: Let X denote the random variable that gives the sum of the faces that fall
uppermost when two fair dice are cast. Find the probability distribution of X.
Example 6: The following data give the number of cars observed waiting in line at the
beginning of 2-minute intervals between 3 and 5 p.m. on a certain Friday at the drive-in
teller of West wood Savings Bank and the corresponding frequency of occurrence. Find
the probability distribution of the random variable X, where X denotes the number of cars
observed waiting in line.
Cars
0
1
2
3
4
5
6
7
8
Frequency of Occurrence
2
8
16
10
9
6
5
3
1
Histograms
A probability distribution of a random variable may be exhibited graphically by means of a
histogram.
In a histogram, the area of a rectangle associated with a value of a random variable X
gives precisely the probability associated with the value of X. This follows because each
such rectangle, by construction, has width 1 and height corresponding to the probability
associated with the value of the random variable. Another consequence arising from the
method of construction of a histogram is that the probability associated with more than
one value of the random variable X is given by the sum of the areas of the rectangles
associated with those values of X.
Example 7: Suppose the probability distribution of a random variable X is represented by
the histogram shown in. Identify that part of the histogram whose area gives the
probability P(10  x  20). Do not evaluate the result.