Download Math 104 ACTIVITY 9: Random variables and probability

Math 104
ACTIVITY 9: Random variables and probability distributions
Why
The language usually used for advanced study in probability, and the language most used in statistics, is that
of random variables and probability distributions.These allow us to extend and organize out study and use of
probability in more situations, and also provide a more compact notation for many events.
LEARNING OBJECTIVES
1. Be able to work effectively as a team, using the team roles
2. Be able to interpret the probability density function for a (discrete - the only type in this course) random variable.
3. Be able to give the probability density function of a discrete random variable from a description of the experiment
involved and the quantity to be observed.
CRITERIA
1. Success in working as a team and in fulfilling the team roles.
2. Success in involving all members of the team in the conversation.
3. Understanding the ideas of a random variable and probability function
RESOURCES
1. The course syllabus
2. The team role desk markers (handed out in class for use during the semester)
3. Your text - Section 4.1
4. 50 minutes
PLAN
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3.
2. Work through the exercises given here - be sure everyone understands all results & procedures.
3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports including team
grade.
DISCUSSION
A random variable is an assignment of a number to each outcome in the sample space of an experiment (a
function defined on the sample space. Random variables are typically named with capital letters (X, Y etc.).
For example, if we roll a die four times, we might have random variables
X = total of the numbers shown
Y = difference between the largest and smallest numbers shown
M = number of times we see a “5” [This would correspond to a sequence of Bernoulli trials experiment]
For any random variable, we are interested in the set of possible values and the probability of each—these
make up the probability density function or probability distribution for the variable. Finding the probability
density function often amounts to solving a lot of very similar probability problems—one for each possible
value of the variable (see the models and the examples in your text)
The random variable notation has one nice advantage— it allows us to express events (in probability calculations) in a clear form. If X is the sum of numbers on two dice, and we want the probability that the sum is
2, we can write P r(X = 2); we likewise write P r(X ≤ 5) for the probability that the sum is no more than 5,
etc.—clearer and simpler than most other notations.
In the next section, we will see how we can discuss the long-term average (the expected value or mean) and
the variation in values (the standard deviation) of a random variable.
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MODELS
1. If we roll a fair die five times and count the number of times we see a “3”, we know how to calculate the probabilities
for various possible values. The variable X = number of “3” ’s is a binomial random variable - because it counts
the number of “successes” in a Bernoulli process. We can give the probability density function either by a formula
or by a table—here we will show both:
The possible values of X are 0, 1, 2, 3, 4, 5. As we know, for each possible value r, the probability is given by
P r(X = r) = C(5, r)( 61 )r ( 56 )5−r
In table form, we have:
x P r(x)
0 C(5, 0)( 16 )0 ( 56 )5 ≈ .4019
1 C(5, 1)( 16 )1 ( 56 )4 ≈ .4019
2 C(5, 2)( 16 )2 ( 56 )3 ≈ .1608
3 C(5, 3)( 16 )3 ( 56 )2 ≈ .0322
4 C(5, 4)( 16 )4 ( 56 )1 ≈ .0032
5 C(5, 5)( 16 )5 ( 56 )0 ≈ .0001
(The decimals don’t quite add up to exactly 1—error in the last place — because of the rounding. If we kept the
exact values (the fractions) they would add to exactly 1)
2. From a group of four women and three men a committee of three people is to be chosen at random. We are interested
in the number of women on the committee. We can let Y represent the number of women on the committee (a
random variable - we can’t know the value until the random process has been carried out). The probability density
function can be given in table form as:
y P r(y)
1
0 C(4,0)C(3,3)
= 35
C(7,3)
1
2
C(4,1)(C(3,2)
= 12
C(7,3)
35
C(4,2)C(3,1)
18
= 35
C(7,3)
C(4,3)C(3,0)
4
= 35
C(7,3)
3
Notice the possible values range from 0 to 3 and the probabilities add to 1.
3. With the situation in the previous model, we could define a variable D to be the difference between the number of
men and the number of women. Possible values of D would be 1 and 3, and the density function would be:
d P r(d)
1 30
35
5
3 35
because D will be 1 when Y is 1 or 2 and D will be 3 when Y is 0 or 3.
EXERCISES
1. A fair die is rolled and a random variable X is defined to be 0 if the number showing is odd, 1 if the number is even.
Give the probability density function of X.
2. A box contains 3 nickels and 4 quarters. Two coins are selected at random from these 7 coins and the random
variable Y is the value of the two coins. Give the probability density function for Y
3. A lottery with 200 tickets sold gives five prizes – a grand prize of $50, two second prizes of $20 each and two third
prizes of $5 each. The tickets for the prizes are drawn at random, without replacement (no one can win two prizes).
We define a random variable V as the amount won by a person who holds one ticket. Give the probability density
function for V .
4. A basketball player makes 30% of her shots. Let the variable X represent the number of shots she makes in a game
in which she takes four shots (assume the shots are independent). Give the probability density function for X
5. From a group of 9 freshmen and 11 sophomores, four students will be selected at random. The random variable Y
represents the number of freshmen selected. Give the probability distribution of Y
READING ASSIGNMENT (in preparation for next class meeting)
Re-read section 4.1 and read section 4.2 in the text
SKILL EXERCISES:(hand in - individually - at next class meeting)
p.151 # 7 – 10, 15 – 16, 18 – 19, 28
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