Probability: a glossary of terms
Preliminaries
A random experiment is an experiment, trial or observation that can
be repeated numerous times under the same conditions, but whose
outcome cannot be predicted with certainty.
An event is a subset of possible outcomes of the random experiment,
which again, may or may not hold for a specific realization of the
random experiment.
The probability of an event is a number between 0 and 1 that
measures the likelihood of the event, i.e., the proportion of times the
event occurs in a large number of repetitions of the random
experiment.
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Random variable and cumulative distribution function
What is a random variable?
A random variable X is a number associated with the outcome of a
random experiment. Its value therefore is not fixed, but can be ascertained
given the outcome of the experiment.
What is a CDF?
A cumulative distribution function (CDF) for a random variable X is the
function
F (x) = Pr(X ≤ x).
Properties of a CDF
A CDF always takes values between 0 and 1.
A CDF is nondecreasing in x.
The limits of a CDF as x approaches −∞ and +∞ are 0 and 1
respectively.
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Working with a CDF
Suppose a random variable X has CDF
F (x) = 1 − e −x ,
0 ≤ x < infty .
Is X more likely to lie in the interval I1 = (ln 2, ln 3] or in I2 = (ln 4, ln 5]?
A. X is more likely to lie in I1
B. X is more likely to lie in I2
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Discrete Random Variable
Definition
A random variable is discrete if it assumes a finite or at most a countably
infinite number of values. If {x1 , x2 , · · · } is the list of possible values of X ,
then pn = Pr (X = xn ) is called the probability density function of X .
Remarks:
A set is countably infinite if the elements in it can be listed in a sequence.
For example, the collection of positive integers is countably infinite, but the
collection of all real numbers between 1 and 2 is not.
Examples of discrete random variables:
I A sample of 100 people is drawn from a population of 100,000. Let X
= number of people in the sample with blood group AB.
I A game consists of rolling a pair of six-faced dice. If the sum of the
faces is 10 or larger, you win $10, else you lose $5. Let X = your
winnings in the game.
I A gardener plants 50 plants on March 1. Let X = number of plants
that have germinated after 15 days.
Math 105 (Section 204)
Probability
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Continuous random variable
What is it?
A random variable is continuous if its cumulative distribution function
F (x) is a continuous function of x. In particular, a continuous random
variable takes an uncountably infinite number of values.
Probability density function
If the cumulative distribution function F is differentiable, the derivative
f (x) = F 0 (x) is called the probability density function of X . In other
words,
Z
x
Pr(X ≤ x) =
f (t) dt.
−∞
Examples:
Height or weight of an individual picked randomly from a population
The waiting time at a bus stop
A number picked from the interval [0, 1].
Math 105 (Section 204)
Probability
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Computing probabilities
Suppose that a pair of fair dice are rolled simultaneously. What is the
probability that the product of the two face values is odd?
A. 1/36
B. 1/4
C. 1/6
D. 3/4
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Probability
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Computing probabilities
Suppose that a pair of fair dice are rolled simultaneously. What is the
probability that the product of the two face values is odd?
A. 1/36
B. 1/4
C. 1/6
D. 3/4
Questions to think about: What is the probability density function of
the product of face values? Graph the cumulative distribution function.
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Working with a PDF
What is the value of k so that
f (x) = kx(1 − x)4 ,
0≤x ≤1
is a probability density function?
A. 1
B. 5
C. 30
D. 6
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