MATH 166 Spring 2016
3.1
c
Wen
Liu
Chapter 3 Probability Distributions and Statistics
3.1 Random Variables and Histograms
Definition: A random variable is a rule that assigns precisely one real number to each outcome of
an experiment.
Types of Random Variables:
1. A random variable is finite discrete if it assumes only a finite number of values.
2. A random variable is infinite discrete if it takes on an infinite number of values that can be
listed in a sequence, so that there is a first one, a second one, a third one, and so on.
3. A random variable is said to be continuous if it can take any of the infinite number of values
in some interval of real numbers.
Example: Classify the following experiments as finite discrete, infinite discrete, or continuous. List
the values of the random variable X.
1. A jelly bean is drawn at random and then replaced from a box of 15 pink and 18 blue jelly
beans. Let the random variable X be the number of draws until a pink jelly bean is picked.
2. X = The number of hours a child watches television on a given day.
3. Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until
an ace is drawn. Let X denote the random variable that gives the number of cards drawn.
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MATH 166 Spring 2016
3.1
c
Wen
Liu
Histograms: Histograms give a vivid description of how the probability is distributed.
Area and Probability: The area of a region of a histogram associated with the random variable
X is equal to P (X), the probability that X occurs. Furthermore the probability that X takes on
the values in the range Xi ≤ X ≤ Xj is the sum of the areas of the histogram from Xi to Xj .
Examples:
1. (p. 121) Suppose a pair of fair sided dice is tossed. Let X denote the random variable that
gives the sum of the top faces.
(a) Find the probability distribution and draw a histogram.
(b) Use the histogram to find the probability P (X = 3), P (X ≤ 3), P (X ≥ 10), and P (9 ≤
X < 12).
2. A box has 4 yellow, 5 gray, and 6 black marbles. Three marbles are drawn at the same time
(i.e. without replacement) from the box. Let X be the number of gray marbles drawn. Find
P (X = 2) and P (X ≤ 2).
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MATH 166 Spring 2016
c
Wen
Liu
3.1
3. An examination consisting of ten true-or-false questions was taken by a class of 100 students.
The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying
histogram. The rectangle with base centered on the number 8 is missing. What should be the
height of this rectangle?
4. A survey was conducted by the Public Housing Authority in a certain community among 1000
families to determine the distribution of families by size. The results are given below.
Find the probability distribution of the random variable X, where X denotes the number of
persons in a randomly chosen family.
Family Size
P (X = x)
2
3
4
5
6
7
8
Binomial Distribution: Given a sequence of n Bernoulli trials with the probability of success p
and the probability of failure 1 − p, the binomial distribution is given by
P (X = k) = C(n, k)pk (1 − p)n−k
for k = 0, . . . , n.
Example: (p. 124) Suppose a fair coin is flipped six times. Let X denote the random variable that
gives the number of “heads”. Find the probability distribution for X.
Number of Successes, x
P (X = x)
0
1
2
3
4
5
6
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