1. Econ 3790: Business and Economics Statistics
Instructor: Yogesh Uppal

2. Lecture Slides 5 Random Variables Probability Distributions
Discrete Distributions
Discrete Uniform Probability Distribution
Binomial Probability Distribution
Continuous Distribution
Normal Distribution

3. Random Variables A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.

4. Example: JSL Appliances
Discrete random variable with a finite number of values
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)

5. Example: JSL Appliances
Discrete random variable with an infinite sequence of values
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.

6. Random Variables Question Random Variable x Type Family size
x = Number of dependents
reported on tax return
Discrete
Distance from
home to store
x = Distance in miles from
home to the store site
Continuous
Own dog
or cat
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
Discrete

7. Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.

8. Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by p(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
p(x) > 0
p(x) = 1

9. Discrete Probability Distributions
Using past data on TV sales, …
a tabular representation of the probability
distribution for TV sales was developed.
Number
Units Sold of Days
200
x p(x)
1.00
80/200

10. Discrete Probability Distributions
Graphical Representation of Probability Distribution
.10
.20
.30
.40
.50
Probability
Values of Random Variable x (TV sales)

11. Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) =  =  p(x) *x
The variance summarizes the variability in the
values of a random variable.
Var(x) =  2 = p(x)*(x - )2
The standard deviation, , is defined as the positive
square root of the variance.

12. expected number of TVs sold in a day
Expected Value
x p(x) x*p(x)
E(x) =
expected number of TVs sold in a day

13. Variance and Standard Deviationx
x - 
(x - )2
p(x)
p(x)*(x - )2 1 2 3 4
-1.2
-0.2
0.8
1.8
2.8
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
.576
.010
.128
.162
.784
Variance of daily sales = s 2 = 1.660
Standard deviation of daily sales = TVs

14. Types of Discrete Probability Distributions:
Uniform
Binomial

15. Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
p(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume

16. Discrete Uniform Probability Distribution
Suppose, instead of looking at the past sales of the TVs, I assume (or think) that TVs sales have a uniform probability distribution, then the example done above would change as follows:

17. expected number of TVs sold in a day
Expected Value
x p(x) x*p(x)
E(x) =
expected number of TVs sold in a day

18. Variance and Standard Deviationx
x - 
(x - )2
p(x)
p(x)*(x - )2 1 2 3 4
-2.0
-1.0
0.0
1.0
2.0
4.0
1.0
0.0 .2
0.8
0.2
0.0
Variance of daily sales = s 2 = 2.0
Standard deviation of daily sales = 1.41 TVs

19. Example: I am bored
Imagine this situation. There is heavy snowstorm. Everything is shut down. You and everybody in your family have to stay home. You are utterly bored. You catch hold of your sibling and get him or her to play this game.
The game is to bet on the toss of a coin.

20. Example: I am bored
If it turns up heads exactly once in three tosses, you win or otherwise you lose.
Lets call the event of getting heads on anyone trial as a success. Similarly, the event of getting tails is a failure.
Suppose the probability of getting heads (or of a success) is 0.6.
The big question is that you want to find out the probability of getting exactly 1 head on three tosses.

21. Tree Diagram Trial 2 Outcomes Trial 1 Trial 3 H
HHH = (0.6)3*(0.4)0= 0.216 H
HHT = (0.6)2*(0.4)1=0.144 T H
HTH = (0.6)2*(0.4)1 =0.144 H T T
HTT = (0.6)1*(0.4)2 =0.096 H
THH = (0.6)2*(0.4)1 = 0.144 H T T
THT = (0.6)1*(0.4)2 =0.096 H T
TTH = (0.6)1*(0.4)2 =0.096 T
TTT = (0.6)0*(0.4)3 =0.064

22. So, what is the probability of you winning the game?
What is the random variable here?
What is the probability of getting 2 heads in three tosses?
= P(HHT) + P(HTH) + P(THH) =
= 0.432
Or 43.2%
How does the probability distribution of our Random variable look?

23. Binomial Distribution
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
4. The trials are independent.

24. Binomial Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
Binomial Distribution is highly useful
when the number of trials is large.

25. Binomial Distribution
Binomial Probability Function
where:
n = the number of trials
p = the probability of success on any one trial

26. Counting Rule for Combinations
Another useful counting rule (esp. when n is large) enables us to count the number of experimental outcomes when x objects are to be selected from a set of N objects.
Number of Combinations of n Objects Taken x at a Time
where: n! = n(n - 1)(n - 2) (2)(1)
x! = x(x - 1)(x- 2) (2)(1)
0! = 1

27. Example: I am bored
Using binomial distribution, the probability of 1 head in 3 tosses is

28. Example: I am bored
Suppose, you won. But knowing your sibling, she or he says that bet was getting exactly 2 heads in 3 tosses. Since you are bored, you have no choice but continuing to play:

29. Example: I am bored
She again cheats. She says that bet was getting at least 2 heads in 3 tosses.
What does this mean: Getting 2 or more heads P(2 heads) + P(3 heads)

30. Example: I am bored

31. Binomial Distribution
Expected Value
E(x) =  = n*p
Variance
Var(x) =  2 = np(1 - p)
Standard Deviation

32. Example: I am bored Mean (or expected value) Variance:
Standard Deviation
E(x) =  = n*p= 3*0.6 = 1.8
Var(x) =  2 = np(1 - p)
= 3*(0.6)*(1-0.6) = 0.72

33. Chapter 6 Continuous Probability Distributions
Normal Probability Distribution x
p(x)
Normal

34. Normal Probability Distribution
The normal probability distribution is the most important distribution for describing a continuous random variable.
It is widely used in statistical inference.

35. Normal Probability Distribution
It has been used in a wide variety of applications:
Heights
of people
Scientific
measurements

36. Normal Probability Distribution
It has been used in a wide variety of applications:
Test
scores
Amounts
of rainfall

37. Normal Distributions
The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the curve between x1 and x2. x
f (x)
Normal x1 x2

38. Normal Probability Distribution
Characteristics
The distribution is symmetric; its skewness
measure is zero. x

39. Normal Probability Distribution
Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode. x
Mean = m

40. Normal Probability Distribution
Characteristics
The entire family of normal probability
distributions is defined by its mean m and its
standard deviation s .
Standard Deviation s x
Mean m

41. Normal Probability Distribution
Characteristics
The mean can be any numerical value: negative,
zero, or positive. The following shows different normal
distributions with different means. x
-10 20

42. Normal Probability Distribution
Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
s = 15
s = 25 x
Same Mean

43. Normal Probability Distribution
Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right). .5 .5 x
Mean m

44. Standardizing the Normal Values or the
z-scores
Z-scores can be calculated as follows:
We can think of z as a measure of the number of
standard deviations x is from .

45. Standard Normal Probability Distribution
A standard normal distribution is a normal distribution with mean of
0 and variance of 1. If x has a normal distribution with mean (μ) and
Variance (σ), then z is said to have a standard normal distribution.
s = 1 z

46. Example: Air Quality
I collected this data on the air quality of various cities as measured by particulate matter index (PMI). A PMI of less than 50 is said to represent good air quality.
The data is available on the class website.
Suppose the distribution of PMI is approximately normal.

47. Example: Air Quality
The mean PMI is 41 and the standard deviation is 20.5.
Suppose I want to find out the probability that air quality is good or what is the probability that PMI is greater than 50.