# STAT 2060 Lecture Notes - Lecture 4: Probability Mass Function, Standard Deviation, Natural Number

180 views10 pages

29 May 2017

School

Department

Course

Professor

STAT*2060: Statistics for Business Decisions

Week 4 Lectures

1 Discrete and Continuous Random Variables

Arandom variable is a variable that takes on numerical values according to a chance process. These

random variables can be discrete or continuous.

Discrete random variables take on a countable, or countably inﬁnite, number of possible values.

That is, for a discrete random variable X, we can (theoretically) count all the possible values Xcan

take on.

In comparison, continuous random variables take on an inﬁnite number of possible values. That

is, for a continuous random variable X, it is not possible to count all of the possible values Xcan take on.

2 Probability Distribution

Very broadly, a probability distribution for a random variable Xdescribes the values Xcan take

on, and the probability associated with each value of X(for a discrete random variable), or with ranges

of values of X(for a continuous random variable).

1

Example: # of heads when flipping a coin 10 times, Height of a randomly selected student

Example: # of heads when we flip a coin 10 times, # of rolls of a die until I roll a 4

We can represent the values of x in a distribution

Visually, x could be 0, 1, 2, 3,4, 5, 6, 7, 88, 9, 10

Examples: Height of randomly selected student, time it takes for randomly selected student to complete a test

Continuous random variable can theoretically take on any value on the real number line

We can represent values of x in a continuous distribution

2.1 Probability Distribution for a Discrete Random Variable

The probability distribution of a discrete random variable Xis a listing of all possible values of X, and

their respective probabilities of occurring.

EXAMPLE: Suppose we conducted an experiment were we ﬂipped two fair coins. Let the random

variable Xbe the number of heads obtained. What is the probability distribution for X?

In order for a discrete probability distribution to be valid, two conditions must be satisﬁed:

1. All probabilities in the distribution must be between 0 and 1.

2. The probabilities in the distribution must sum to 1.

Many of our basic probability rules and calculations can be carried out on a discrete random variable,

by using the probabilities found in the probability distribution.

EXAMPLE: (Adapted from Introduction to Probability and Statistics, 10th Ed. (1999). Mendenhall, W., Beaver, R.J.,

and Beaver, B.M. Duxbury Press. pg162) An electronics store sells a particular model of computer notebook,

with only a limited number of the product in stock on any given day. Let the random variable X

represent the daily demand for the notebook (i.e. the number of customers per day who want to

purchase it), with the following probability distribution:

X012345

P(X) 0.10 0.40 0.20 0.15 0.10 0.05

i) What is the probability that the store will have exactly three customers wanting to purchase the

notebook?

ii) What is the probability that the store will have at least four customers wanting to purchase the

notebook?

2

What values can x take on? 0 ,1, 2

P(X=0)=0.25-Tail & Tail

P(X=1)=0.5

P(X=2)=0.25

Probability distribution

P(X)

X

0

1

2

0.25

0.50

0.25

P(3)=0..20

P(at least 4)=0.10 + 0.05

=0.15

P

iii) What is the probability that the store will have less than four customers wanting to purchase the

notebook?

iv) What is the probability that the store will have at least four customers wanting to purchase the

notebook, given that they have already sold at least three notebooks?

2.2 Expected Value and Variance of a Random Variable

For any discrete random variable X, we can calculate the expected value and the variance using the

probability distribution.

The expected value, denoted by E(X) or sometimes just µ, is the theoretical mean of a random variable.

As such, the expected value is a parameter.

We calculate the expected value for a random variable Xas:

E(X) = µ=X

all x

xP (x)

EXAMPLE: Return to our probability distribution for the random variable X, where Xis the number

of heads attained when two fair coins are ﬂipped.

X012

P(X) 0.25 0.50 0.25

3

P(less than 4)=0.85

P(at least 4 I sold at least 3 notebooks)=P(AnB)/P(B)

= 0.15/0.30

=0.50

P(A)=X greater than or equal to 4

P(B)=X greater than or equal to 3

P(AnB)=X=4, X=5

If we were to repeat our experiment an infinite number of times, the expected value is the theoretical mean value of x that we

would get

Summing up all values of x & multiply by their respected probabilities

E(X)=1(0.5) +2(0.25)

=1.00

What is expected value of X, if we were too flip 2 fair coins an infinite amount of times