Lecture Note 5

1
Lecture Note 5

• Discrete Random Variables and Probability
  Distributions

2
Random Variables

• A random variable is a variable that takes on
  numerical values determined by the outcome of a
  random experiment.

3
Discrete Random Variables

• A random variable is discrete if it can take on
  no more than a countable number of values.

4
Discrete Random Variables(Examples)

• The outcome of a roll of a die.
• The number of defective items in a sample of
  twenty items taken from a large shipment.

5
Continuous Random Variables

• A random variable is continuous if it can take
  any value in an interval.

6
Continuous Random Variables(Examples)

• The income in a year for a family.
• The amount of oil imported into the U.S. in a
  particular month.

7
Discrete Random Variables

• We will study the fundamental concepts of
  discrete random variables, namely (1) Probability
  Distribution Function and (2) Cumulative
  Probability Function.
• First, we will learn the notation of the
  Probability Distribution Function

8
Understanding the notation for the probability
distribution function

• Suppose you are about to roll a die. Then, number
  you would get after the roll is a discrete random
  variable. Let X denote this random variable.
• Then this random variable X can take 6 possible
  outcomes from S1, 2, 3, 4, 5, 6.

9
Understanding the notation for the probability
distribution function, contd

• Then, probability that the random variable X
  takes the value 1 (i.e., the probability that you
  get 1 after rolling the die) is 1/6. This can be
  conveniently expressed using the following
  notation.
• P(1)P(X1)1/6
• Similary, we have
• P(2)P(X2)1/6
• P(3)P(X3)1/6
• .
• .
• P(6)P(X6)1/6

10
Understanding the notation for the probability
distribution function, contd

• We can generalize the notation in the previous
  slides to other discrete random variables. let X
  be any discrete random variable, which takes k
  possible values from Sx1, x2, , xk
• Conventionally, we use capital letter X to
  denotes the random variable, and small letter x
  to denote the possible values that X can take.

11
Understanding the notation for the probability
distribution function, contd

• Now, let x be any values from Sx1, x2, , xk
• Then the probability that the random varialbe X
  takes the value x can be written in the following
  notation
• P(x)P(Xx)
• This notation is what is called Probability
  Distribution Function. The next slide re-iterate
  this definition.

12
Probability Distribution Function

• The probability distribution function, P(x), of a
  discrete random variable expresses the
  probability that X takes the value x, as a
  function of x. That is

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Probability Distribution Function-Exercise 1-

• Ex 1-1 Suppose you roll a die. Let X be the
  random variable for this experiment. Find the
  probability distribution function P(x) by
  completing the table in the Excel Sheet
  Probability Distribution Function Exercise.
• EX 1-2 Plot the probability distribution
  function.

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Probability Distribution Function-Exercise 2-

• Consider the following game. You toss a coin. If
  you get the heads, you receive \100. If you gets
  the tales, you receive none.
• The payoff for this game is a discrete random
  variable. Let X denotes this random variable.
  Find the probability distribution of this random
  variable, by completing the table in Probability
  Distribution Function Exercise

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Required Properties of Probability Distribution
Functions of Discrete Random Variables

• Let X be a discrete random variable with
  probability distribution function, P(x). Then
• P(x) ? 0 for any value of x
• The individual probabilities sum to 1 that is
• Where the notation indicates summation over all
  possible values x.

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Cumulative Probability Function

• The cumulative probability function, F(x0), of a
  random variable X expresses the probability that
  X does not exceed the value x0, as a function of
  x0. That is
• Where the function is evaluated at all values x0

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Cumulative Probability Function-Example 3-

• 3-1 Let X be the random variable for a roll of a
  die. Find F(1), F(2), F(3), to F(6).
• 3-2 Plot F(x) and x.

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Derived Relationship Between Probability Function
and Cumulative Probability Function

• Let X be a random variable with probability
  function P(x) and cumulative probability function
  F(x0). Then it can be shown that
• Where the notation implies that summation is over
  all possible values x that are less than or equal
  to x0.

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Expected Value

• Expected value is a similar concept as the
  average. More specifically, it is the weighted
  average with the weight given by the probability
  of each outcome.
• For example, consider the game. You toss a coin.
  If you get the heads, you receive \100. If you
  get the tails, you receive none.
• Then the expected payoff for this game is
• \100(probability of getting \100)
• \0(probability of getting \0) \50
• More formally, expected value is defined as
  follow. See next slide.

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Expected Value

• The expected value, E(X), of a discrete random
  variable X is defined
• Where the notation indicates that summation
  extends over all possible values x.
• The expected value of a random variable is called
  its mean and is denoted ?x.

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Understanding the notation of the expected value

• Let X be the random variable which takes
  value in Sx1, x2,..,xk.
• Then the expected value of X is computed as
  follows. See next slide

22
µX
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Expected Value Exercise-Exercise 4-

• Let X be the random variable that shows the the
  number of house purchase contracts that an real
  estate agent can achieve in a month. The
  probability distribution of X is given by the
  following.
• Compute the expected number of house
  purchase contracts that the real estate agent can
  achieve in a month.

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Expected Value Functions of Random Variables

• Let X be a discrete random variable with
  probability function P(x) and let g(X) be some
  function of X. Then the expected value, Eg(X),
  of that function is defined as

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Expected Value Functions of Random
Variables-Exercise 5-

• Continue using the real estate agent example.
  Suppose that the monthly salary for the real
  estate agent is given by
• g(x) 1000 1500x
• Ex 5-1 Find the expected monthly salary of
  the agent. Use the Probability Distribution
  Function Exercise Excel Sheet.

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Variance and Standard Deviation

• Let X be a discrete random variable. The
  expectation of the squared discrepancies about
  the mean, (X - ?)2, is called the variance,
  denoted ?2x and is given by
• The standard deviation, ?x , is the positive
  square root of the variance.
• Often we use the Var(X) to denote the variance
  and SD(X) to denote the standard deviation

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Understanding the procedure to compute the
variance

• The table in the next slide summarizes the
  procedure to compute variance.

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Variance and Standard Deviation-Exercise 6-

• Continue using the real estate agent example.
  Using the table in the Probability Distribution
  Function Exercises Excel sheet, answer the
  following questions.
• Ex 6-1 Compute the variance
• Ex 6-2 Compute the standard deviation

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Summary of Properties for Linear Function of a
Random Variable

• Let X be a random variable with mean ?x , and
  variance ?2x and let a and b be any constant
  fixed numbers. Define the random variable Y a
  bX. Then, the mean and variance of Y are
• and
• so that the standard deviation of Y is

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Linear Function of a Random Variable-Exercise 7-

• Continue using the example of real estate agent
  from exercise 4, 5, and 6.
• Suppose that the monthly salary for this agent,
  Y, is determined in the following method.
• Y1000 1500X
• Ex 6-1 Compute E(Y) using the formula in the
  previous slide.
• Ex 6-2 Compute the variance and standard
  deviation of Y

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Standardization of a Random Variable

• Let X be a random variable with mean ?X and
  standard deviation ?X. Define a new random
  variable Z as
• Then
• E(Z)0
• Var(Z)1

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Exercise 8

• Continue using the real estate agent example. Let
  X be the random variable for the possible number
  of contracts the agent can achieve in a month.
• Standardize X using the formula in the previous
  slide, and verify that this is mean zero and
  variance 1.

34
Reviews of Combinatorics and Common discrete
distributions

• Review of Combinatorics
• Number of orderings
• Number of combinations
• Number of sequences of x Successes in n Trials
• Common discrete distributions
• Bernoulli Distribution
• Binomial Distribution

35
Stock Price Movement Example

• To motivate the study of combinatrics and the
  discrete distributions, let us consider a simple
  example of stock price movement. The example will
  utilize these combinatrics and distributions.

36
Stock Price Movement Example

• The price of Stock A today is 10. At the end of
  each month, the price of the stock A either goes
  up by the factor of 1.2 with probability 0.4, or
  goes down by the factor of 0.9 with probability
  0.6.
• This means that, if the price of the stock goes
  up this month, then the price of the stock in the
  second month will be 101.2 12. If the price
  of the stock goes up this month, and goes down in
  the second month, the price of the stock in the
  third month will be 101.20.9 10.8.

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Stock Price Movement Example, Contd

• Suppose that you make the following contract with
  a stock broker.
• You have the right to purchase a share of
  stock A at the price equal to 37.61 at the
  beginning of the 13th month (That is, a year from
  today).
• Then, if the actual market price of stock A is
  higher than 37.61, you purchase the stock and
  immediately sell it to make a positive profit. If
  the actual price is lower than 37.61, you do not
  have to exercise the right.
• Then consider the following question.
• See next slide

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Stock Price Movement Example, Contd

• Question
• What is the probability that you make some
  strictly positive profit from this contract
  (ignoring all the fees associated with the
  contract and buying and selling of the share)
• Answer to this question requires tools such as
  combinatorics and Binomial distributions.
• In the following slides, we will review
  combinatorics and some common distributions.
  After studying these concepts, we will come back
  to the question again.

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Review of Combinatorics-Number of orderings-

• You have n cards numbered from 1 to n. The number
  of ways you can order this card is given by
• n!n(n-1) (n-2) 321
• n! reads n factorial
• 0! is defined to be 1.

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Number of ordering example

• There are 5 cards numbered from 1 to 5. What is
  the number of ways you can order this card?

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Review of Combinatorics-Combinations-

• Suppose there are n cards numbered from 1 to n.
  If you take x cards out of n cards, the number of
  possible combinations of the cards is given by

Where n!n(n-1) (n-2) 321
Cnx reads n choose x.
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Combinations -Example-

• A personnel officer has eight candidate to fill
  four similar positions. What is the total number
  of possible combinations of four candidate chosen
  from eight?

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Review of combinatorics- sequences of x
Successes in n Traials-

• Consider a random trial where there are only two
  outcomes, Success or Failure.
• Then, if you make n independent trials, the
  number of sequences that contains exactly x
  successes is equal to Cnx.

44
Sequences of Successes-Exercise-

• Consider tossing a coin 10 times. Find the number
  of ways the head appears exactly 4 times.

45
Number of sequences-Exercise-

• Suppose that the figure below is a road map of a
  certain area. If you start from point A and walk
  to point B, how many possible routes can you
  take? (Suppose you do not walk back)

B
A
46
Bernouli Trial

• Bernouli Random Trial with success probability ?.

• This is a random experiment with two possible
  outcomes, success or failure, where the
  probabilities are given by
• P(Success) ?
• P(Failure) 1- ?

47
Bernouli Random Variable

• Consider a Bernouli Trial with P(Success) ?.
  Define a random variable X in the following way.
• X1 if the Bernouli random trial turn out to be
  success
• X0 if the Bernouli random trial turns out to be
  a failure.
• Then, X is called a Bernouli Random Variable
  with success probability ?.

48
Bernouli Distribution

• Consider a Bernouli random variable X with
  success probability ?. Then probability
  distribution function for X is called Bernouli
  Distribution with success probability ?. This is
  given by
• P(0)(1- ?) and P(1) ?

49
Bernouli Distribution -Example-

• Consider the following game. You toss a coin. If
  you get the heads, you receive 1. If you get the
  tails, you receive none. Let X be the random
  variable for the payoff of this game.
• X has the Bernouli distribution with success
  probability 0.5.

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Mean and Variance of a Bernoulli Random Variable

• The mean is
• And the variance is

51
Binomial Distribution-Example-

• I use an example to illustrate Binominal
  Distribution
• You inspect a production line by randomly
  checking the items. It is known that 5 of the
  products from this production line are defect
  items.
• Suppose you randomly choose 4 items from the
  production line. Then number of defect items you
  would find is a discrete random variable.

52
Binomial Probability -Examples, Contd-

• Let X be the number of defect items in the 4
  randomly picked items. What is the probability
  that exactly 2 of them are defect items?
• To answer to this question, first, consider the
  possible sequences that 2 defect items are
  picked.

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• The number of possible sequences you pick 2
  defect items is given in the table. D denote the
  defect item, and G denote the good item.
• Note that the number of sequences is given by
  C42.

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• Since the probability of picking one defect item
  is 0.05, and the probability of picking one good
  item is 0.95, the probability of getting each
  sequence is 0.0520.9520.002256.
• Since there are 6 sequences, the probability that
  you get exactly 2 defect items is given by
  (0.002256)60.0135

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• We can generalize this problem. Consider a
  production line. The probability that an item
  from this production line is a defect item ?.
• Suppose that you pick n products from this
  production line. Then, the probability that the
  number of defect items is exactly x is given by

• This probability distribution function is called
  Binomial Distribution with success probability
  ?. Next slide summarizes the binomial
  distribution.

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Binomial Distribution

• Suppose that a random experiment can result in
  two possible mutually exclusive and collectively
  exhaustive outcomes, success and failure, and
  that ? is the probability of a success resulting
  in a single trial. If n independent trials are
  carried out, the distribution of the resulting
  number of successes x is called the binomial
  distribution. Its probability distribution
  function for the binomial random variable X x
  is
• P(x successes in n independent trials)
• for x 0, 1, 2 . . . , n

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Mean and Variance of a Binomial Probability
Distribution

• Let X be the number of successes in n independent
  trials, each with probability of success ?. The
  x follows a binomial distribution with mean,
• and variance,

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Binomial Probabilities- An Example
A sales person randomly visits houses to sell a
certain product. He believes that for each visit,
the probability of making a sale is 0.40.
If the sales person visits 5 houses, what is the
probability that he makes at least 3 sale?
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Answer
Let X be the random number for the number of
sales. Then, P(At leaset 3 sale) P(X 3)
P(X 3) P(X 4)P(X5)
P(makes at least 3 sales)P(3)P(4)P(5) 0.23040
.07680.010240.31744
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Computing Binomial Distribution using Excel

• Let n be the total number of trial. Let x be
  the number of success, and ? be the success
  probability. Excel function to compute binimial
  probabilities is
• P(Xx) BINOMDIST(x, n, ?, FALSE)
• P(Xx)BINOMDIST(x,n, ?, TRUE)
• Note that if you put FALSE at the end, it
  computes the binomial probability distribution.
  If you put TRUE at the end, it computes the
  cumulative binomial distribution.

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Binomial Distribution-Exercise-

• Open Binomial Distribution Exercise
• Find the Binomial Distribution Function for n50
  and ?0.3. Then, graph the Binomial Probability
  Distribution.

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Binomial Distribution Function with n50 and
?0.3.
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Stock Price Movement Example

• Now, we come back to the Stock Price Movement
  example that we saw at the beginning.
• First, take a look at how the stock price may
  move in the first three months. See the next slide

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Stock price movement for the first three months
As can be seen, there are three possible values
for the stock price in the third month. Now
answer the questions in the next slide.
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• Q1 In the 13th month, how many possible values
  for the stock price are there?
• Q2. Show that the 4th highest price is 37.61
• Q3. What is the probability that the stock price
  at the beginning of 13th month is equal to the
  4th highest price, 37.61?
• Q4. Remember the contract. The contract is You
  have the right to purchase a share of the stock A
  in the 13th month at the price equal to 37.61.
  Then what is the probability that you get some
  positive profit from this contract (ignoring all
  the fees associated with buying and selling the
  stock)

66
Joint Probability Functions

• Consider two stocks Stock A and Stock B. Let X
  denote the random variable for the return of
  stock A. Let Y denote the random variable for the
  return of Stock B.
• Suppose that X takes 4 possible values 0, 0.05,
  0.1, and 0.15.
• Further, suppose that Y takes 4 possible values,
  0, 0.05, 0.1 , and 0.15.
• The joint probabilities of X and Y are given in
  the table in the next slide.

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Joint Probability Function-Example-
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Joint Probability Functions

• Let X and Y be a pair of discrete random
  variables. Their joint probability function
  expresses the probability that X takes the
  specific value x and simultaneously Y takes the
  value y, as a function of x and y. The notation
  used is P(x, y) so,

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Marginal Probability Functions

• Let X and Y be a pair of jointly distributed
  random variables. In this context the
  probability function of the random variable X is
  called its marginal probability function and is
  obtained by summing the joint probabilities over
  all possible values that is,
• Similarly, the marginal probability function of
  the random variable Y is

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Exercise

• Open Joint Probability Exercise. This sheet
  contains the joint probability example of stock A
  return (X) and Stock B return (Y) in the previous
  example.
• Find the Marginal distribution.

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Conditional Probability Functions

• Let X and Y be a pair of jointly distributed
  discrete random variables. The conditional
  probability function of the random variable Y,
  given that the random variable X takes the value
  x, expresses the probability that Y takes the
  value y, as a function of y, when the value x is
  specified for X. This is denoted P(yx), and so
  by the definition of conditional probability
• Similarly, the conditional probability function
  of X, given Y y is

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Exercise

• Continue using the same example. Compute the
  following conditional probabilities.
• P(X0Y0.1)
• P(Y0.15X0.1)

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Independence of Jointly Distributed Random
Variables

• The jointly distributed random variables X and Y
  are said to be independent if and only if their
  joint probability function is the product of
  their marginal probability functions, that is, if
  and only if
• And k random variables are independent if and
  only if

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Exercise

• Using the example in the Joint Distribution
  Exercise, check to see if X and Y are
  statistically independent.

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Covariance

• Let X be a random variable with mean ?X , and let
  Y be a random variable with mean, ?Y . The
  expected value of (X - ?X )(Y - ?Y ) is called
  the covariance between X and Y, denoted Cov(X,
  Y).
• For discrete random variables
• An equivalent expression is

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Correlation

• Let X and Y be jointly distributed random
  variables. The correlation between X and Y is

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Covariance and Statistical Independence

• If two random variables are statistically
  independent, the covariance between them is 0.
  However, the converse is not necessarily true.

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Exercise

• Consider the following joint distribution of X
  and Y

Exercise Cov(X,Y), Var(X), Var(Y) and CORR(X,Y).
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Portfolio Analysis

• The random variable X is the price for stock A
  and the random variable Y is the price for stock
  B. The market value, W, for the portfolio is
  given by the linear function,
• Where, a, is the number of shares of stock A and,
  b, is the number of shares of stock B.

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Portfolio Analysis

• The mean value for W is,
• The variance for W is,
• or using the correlation,

81
Exercise

• Consider Stock A and Stock B. Let X and Y denote
  the market price of stock A and B respectively.
  It is known that E(X)10, E(Y)20, Var(X)2,
  Var(Y)4, and Cov(X,Y)?1. Now consider the
  following portfolio.
• W5X 4Y.
• Find E(W), Var(W).