1305AFE Lecture Notes - Lecture 5: Standard Deviation, Normal Distribution, Probability Density Function
Week 5 Business Data Analysis Lecture Notes
Probability and Probability Distribution
Random Variables
• A random variable is a function that assigns a numerical value to each possible
outcome of an experiment
• Example:
o Experiment: Tossing two coins
o Sample space: {HH, HT, TH,TT}
o If X is a random variable that counts the number of heads occurring from this
experiment,
o X= {0 , 1 , 2 }
Two Types of Random Variables
• Discrete random variable
o One that takes on a countable number of values
o Example:
▪ 1. X= Number of students coming to a class on a given day
={,,,….}
▪ 2. Y= Number of prawns caught by a fisherman on a given week
▪ ={,,,….}
▪ 3. Z= odd ubers draw fro a lotto ={,,,7….}
• Continuous random variable
o Variables whose values are not discrete, not countable
o Example:
▪ Time, weight, height, length, income, profit, etc.
Discrete and continuous Random Variables
• A random variable is discrete if it can assume only a countable number of values
o Discrete random variable
▪ After the first value is defined the second value, and any value
thereafter, are known.
▪ Therefore, the number of values is countable
• A random variable is continuous if it can assume an unaccountably infinite number
of values
o Continuous random variable
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▪ After the first value is defined, any number can be the next one
▪ Therefore, the number of values is unaccountably infinite
Discrete Probability Distribution
• The probability that the random variable X will equal x is: P(X=x), or more simply p(x)
• To calculate P(X=x), the probability that the random variable X assumes the value x,
add the probabilities of all the simple events for which X is equal to x
Requirements of a Discrete Probability Distribution
• If a random variable can take values X=xi, then the following must be true:
Probabilities as Relative Frequencies
• In practice, probabilities are often estimated from relative frequencies
• Example:
o The number of cars a dealer is selling daily was recorded over
the last 200 days. The data are summarized as follows:
▪ Estimate the probability distribution
▪ State the probability of selling more than 2 cars a day
• Solution:
o From the table of frequencies we can calculate the relative
frequency, which becomes our estimated probability distribution
1)x(p.2
xallfor1)p(x0 .1
i
xall
i
ii
=
££
å
Daily sales Frequency
010
130
270
350
440
200
Daily sales Relative frequency
010/200 = 0.05
130/200 = 0.15
270/200 = 0.35
350/200 = 0.25
440/200 = 0.20
1.00
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Calculating Mean µ or Expected Value E(X) and Variance σ2 of Discrete Probability
Distribution
• The discrete probability distribution represents a population
• Therefore, we calculate the population parameters
• E.g. the population mean and population variance
The Population Mean (or Expected Value)
• Population mean is also known as the expected value of X, denoted by E(X).
• Given a discrete random variable X with values xi, i=1,2,3,..,k that occur with
probabilities p(xi), the expected value of X is
Variance
• The population variance is calculated in a similar manner
• Let X be a discrete random variable with possible values Xi that occur with
probabilities p(xi), i=1,2,3,..,k and let the mean E(X)= . the Variance of X is defined
to be
• Alternatively,
Standard Deviation
• the standard deviation of a random variable X, denoted Ó, is the positive square root
of the variance of X
• Example:
o The total number of cars to be sold next week is described by the following
probability distribution
o 1. Find the mean or the expected number of cars the dealer will
sell in a week
o 2. Find the Variance of cars the dealer will sell in a week.
11 2 2
( ) ( ) .....
i
ii k k
all x
EX x p x x p x p x p
µ
== ×= + + +
å
22
() ( ) ( )
i
ii
all x
VX x p x
sµ
== -
å
22 2
1()
i
i
all x
xpx
sµ
= -
å
( ) VAR( )
SD X X
s
==
xip(xi)
0 0.05
1 0.15
2 0.35
3 0.25
4 0.20
1.00
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Document Summary
Random variables: a random variable is a function that assigns a numerical value to each possible outcome of an experiment, example, experiment: tossing two coins, sample space: {hh, ht, th,tt} If x is a random variable that counts the number of heads occurring from this experiment: x= {0 , 1 , 2 } Two types of random variables: discrete random variable, one that takes on a countable number of values, example, 1. X= number of students coming to a class on a given day: continuous random variable, variables whose values are not discrete, not countable, example, time, weight, height, length, income, profit, etc. Y= number of prawns caught by a fisherman on a given week: 3. ={(cid:1004),(cid:1005),(cid:1006), . (cid:1009)(cid:1004)(cid:1004)(cid:1004): after the first value is defined, any number can be the next one, therefore, the number of values is unaccountably infinite. If a random variable can take values x=xi, then the following must be true: for1) i xall i.
