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Chapter 5
Random Variables
5.1 Random Variables
A random
variable is an assignment of numbers to outcomes of a random
experiment. For example, consider the experiment of drawing tickets at random
independently from a box of numbered tickets. The possible outcomes of n
such draws are sequences of n tickets in a particular order. The sample sum
of the numbers on the tickets drawn is a random variable. Depending on the
sequence of tickets drawn, the sample sum takes different values. To any
particular sequence of n tickets, the sample sum assigns the sum of
their labels.
For discrete probability
distributions, the number of values that have nonzero probability is countable.
The binomial probability distribution assigns positive probability to the
integers {0, 1, . . . , n}.
There are many other
discrete distributions. Some have names; some do not. presents a random
variable with a discrete distribution that has no name (as far as I am aware).
random variable
is an assignment of numbers to outcomes of a random experiment. If a random
variable has a countable number of possible values, it is discrete.
The probability
distribution of a random variable says, for each possible value of
the random variable, what the chance is that the random variable will equal
that value. Probability distributions can be given by formulae or tables. If a
probability distribution assigns probability 100% to the union of a countable
set of outcomes, the probability distribution is said to be discrete.
Probability distributions of discrete random variables are discrete.
Consider a box of N
tickets of which G
are labeled "1" and N-G are labeled "0." The sample sum of
the labels on n
tickets drawn at random with replacement from the box has a binomial
distribution with parameters n and p=G/N;
the probability that the sample sum equals k is
nCk
× pk
×(1-p)n-k,
for k=0,
1, . . . , n.
Random
variables describe selected features of elements of a sample space. In this
part of the course, we are only interested in quantitative random variables.
From a mathematical point of view, a (quantitative) random variable is a function
that assigns a numerical value to each element of a sample space. They come in
two kinds: discrete and continuous random variables. A continuous random
variable can take any value in an interval or in several intervals of real
numbers, whereas in the case of a discrete random variable there are gaps
between consecutive possible values. While some random variables are naturally
discrete (for example, the number of siblings a person has), in many cases it
is a matter of choice whether a given feature should be modeled as a discrete
or a continuous random variable. For example, physical entities such as height,
weight, temperature, pressure, etc. are "naturally" continuous random
variables. But if we decide to measure, say, height of newborn humans in inches,
then we will have very few possible integer values of this variable, and it is
more meaningful to treat it as a discrete random variable.
Let
T be the time that elapses until a given atom of a radioactive substance
disintegrates. Is T a continuous or a discrete random variable?
ˇ
continuous
ˇ
discrete
Let
T be the number of days in a given quarter when classes are in session (that
is, excluding holidays and such). Is T a continuous or a discrete random
variable?
ˇ
continuous
ˇ
discrete
For the rest of this tutorial, we will
concentrate on discrete random variables. The PROBABILITY DISTRIBUTION of a
random variable indicates how likely which values of the random variable are.
The probability distribution of a discrete random variable can be represented by
the PROBABILITY FUNCTION of this random variable. The probability function
assigns to each possible value of a random variable its probability. For
example, if the experiment consists of tossing a fair coin twice, and if X is
the random variable that counts the number of heads that come up, then X can
take on the values 0, 1, 2 and the probability function of X is given as
p(0)=0.25, p(1)=0.5, p(2)=0.25.
Suppose your
experiment consists of tossing a biased coin twice. On a single toss, the coin
will come up heads with probability 0.6 and tails with probability 0.4. Let X
be the random variable that counts the number of times heads comes up. Which of
the following is the probability function of X?
ˇ
p(0.36)=2, p(0.48)=1,
p(0.16)=0
ˇ
p(0.16)=2, p(0.48)=1,
p(0.36)=0
ˇ
p(2)=0.36, p(1)=0.48,
p(0)=0.16
ˇ
p(0)=0.36, p(1)=0.48,
p(2)=0.16
Probability functions are frequently
given in the form of tables. For example, the probability function of the
previous question can be represeted by the following table:
|
x |
0 |
1 |
2 |
|
p(x) |
0.16 |
0.48 |
0.36 |
Let us consider a somewhat more involved example.
Roll
a fair die twice. Let X be the number that comes up on the first roll, and let
Y be the number that comes up on the second roll. Let Z = (X - Y)2.
Which of the following is the probability function of Z?
ˇ
p(X=1)=1/6,
p(X=2)=1/6, p(X=3)=1/6, p(X=4)=1/6, p(X=5)=1/6, p(X=6)=1/6
ˇ
p(Z=0)=1/6,
p(Z=1)=1/6, p(Z=2)=1/6, p(Z=3)=1/6, p(Z=4)=1/6, p(Z=5)=1/6
ˇ
p(Z=0)=1/6,
p(Z=1)=1/6, p(Z=4)=1/6, p(Z=9)=1/6, p(Z=16)=1/6, p(Z=25)=1/6
ˇ
p(Z=0)=1/6, p(Z=1)=10/36,
p(Z=4)=8/36, p(Z=9)=6/36, p(Z=16)=4/36, p(Z=25)=2/36
ˇ
p(Z=0)=1/6,
p(Z=1)=10/6, p(Z=4)=8/6, p(Z=9)=6/6, p(Z=16)=4/6, p(Z=25)=2/6
ˇ
p(Z=0)=1/36,
p(Z=1)=1/36, p(Z=2)=1/36, p(Z=3)=1/36, p(Z=4)=1/36, p(Z=5)=1/36
An important feature
of probability functions is that the values it takes are probabilities that add
up to 1. This fact often allows one to determine missing values.
A
random variable X takes the integer values 1 through 4. If p(X=1) = 0.1 and
p(X=2)=p(X=3)=p(X=4), find p(X=4).
ˇ 0.1
ˇ
0.2
ˇ
0.3
ˇ
0.4
ˇ
0.5
ˇ
0.6
ˇ
0.7
ˇ
0.8
ˇ
0.9
Many random variables
that occur in various applications have a binomial distribution. Such a
distribution occurs whenever a random variable can be conceptualized as
counting the number of "successes" in several independent
"trials" of an experiment where the probability of success in each
trial is the same. The probability of x successes in n trials with probability
p of success in each individual trial is often denoted by b(x;n,p).
Questions that
involve the binomial distribution can be solved by following a fixed sequence
of steps.
1. Determine
what the "trials" are and what "success" in a single trial
means. Remember that "success" is a purely mathematical concept and
does not need to correspond to a desirable outcome.
2. Determine
the number n of trials.
3. Determine
the probability p of success in a single trial.
4. Reformulate
the question in terms of b(x;n,p)'s. This step will usually include determining
what the relevant x (or x's) is (or are).
5. Find
the numerical answer using tables, the formulas given in your textbook, or a
calculator. This last step is often the least problematic and will not be
discussed in this tutorial.
On a multiple choice
test, there are ten questions with five possible answers each. Dan knows the
correct answers to five of these questions and randomly guesses the answers for
the remaining questions. What is the probability that Dan gives exactly eight
correct answers on this test?
ˇ
b(8;0.5,10)
ˇ
b(8;10,0.5)
ˇ
b(8;0.2,10)
ˇ
b(8;10,0.2)
ˇ
b(3;0.5,5)
ˇ
b(3;5,0.5)
ˇ
b(3;0.2,10)
ˇ
b(3;10,0.2)
ˇ
b(3;0.2,5)
ˇ
b(3;5,0.2)
Jane
cought the flu on campus in Athens. She goes home for the weekend. Each of her
five younger siblings has a 40% chance of catching the flu from Jane over the
weekend. What is the probability that Jane will infect at most one of her
younger siblings?
ˇ
b(1;5,0.4)
ˇ
b(0;5,0.4) +
b(1;5,0.4)
ˇ
1-(b(0;5,0.4) +
b(1;5,0.4))
ˇ
b(0;5,0.6) +
b(1;5,0.6)
A
fair coin is tossed eight times. What is the probability that heads comes up
more often than tails?
Good
luck on the quiz!!!