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

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29 May 2017
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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
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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
0.50
P(3)=0..20
P(at least 4)=0.10 + 0.05
=0.15
P
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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
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