1. Probability Distributions Random Variables: Finite and Continuous Distribution Functions Expected value
April 3 – 10, 2003

2. Random Variables
A random variable is a rule that assigns a numerical value to each outcome of an experiment
Two types:
Discrete:
Finite: It can take on only finitely many possible values (ex: X=0,1,2, or 3). In this case you can list all possible values.
Infinite: It can take on infinitely many values that can be arranged in a sequence (ex: X=1,2,3,4,…)
Continuous: If the possible values form an entire interval of numbers (ex: any positive number)

3. Discrete Random Variables
We want to associate probabilities with the values that the random variable takes on.
There are two types of functions that allow us to do this:
Probability Mass Functions (p.m.f)
Cumulative Distribution Functions (c.d.f)

4. Probability Distributions
The pattern of probabilities for a random variable is called its probability distribution.
In the case of a finite random variable we call this the probability mass function (p.m.f.), fx(x) where fx(x) = P( X = x )

5. Probability Mass Function (p.m.f)
Using the p.m.f. we can describe various probabilities of X geometrically
EX: Let X describe the number of heads obtained when you toss a fair coin twice. x 1 2
P(X=x) or fX
.25
.50
From this table, we have the ordered pairs (0,.25),(1,.5),(2,.25)

6. Probability Mass Function
This is a p.m.f which is a histogram representing the probabilities
When a histogram is used, the r.v. X takes on integer values
In this case P(X=x) equals the area of the rectangle
Note: For a histogram to represent a p.m.f, the heights of the rectangles should sum to 1
This is because the values along the y-axis represent probabilities

7. Cumulative Distribution Function
The same probability information is often given in a different form, called the cumulative distribution function (c.d.f) or FX
FX(x) = P(Xx)
0  FX(x)  1, for all x
In the finite case, the graph of a c.d.f. should look like a step function, where the maximum is 1 and the minimum is 0.

8. Cumulative Distribution Function

9. Graphing a CDF (Finite case)
Need to look at every possible x along the x-axis and see what value of the cdf corresponds to it
Look at intervals – i.e, less than 0, between 0 and 1, between 1 and 2, etc.
When looking at intervals, include the left most number but not the right most number – i.e, between 0 and 1, include 0 but not 1
Find the value of FX(x) = P(X  x) that corresponds
For each interval you are looking at, FX(x) should be the same number

10. Bernoulli Trials
In a Bernoulli Trial there are only two outcomes: success or failure
Let p = P(S)
Bernoulli Random Variables were named after Jacob Bernoulli (1654 – 1705) who was a famous Swiss mathematician

11. Binomial Random Variable
Let X stand for the number of successes in n Bernoulli Trials where X is called a Binomial Random Variable
Binomial Setting: 1. You have n repeated trials of an experiment 2. On a single trial, there are only two possible outcomes
3. The probability of success is the same from trial to trial
4. The outcome of each trial is independent
Expected Value of a Binomial R.V is represented by E(X)=n*p

12. BINOMDIST
BINOMDIST is a built-in Excel function that gives values for the p.m.f and c.d.f of any binomial random variable
It is located under Statistical in the Function menu
Syntax:
BINOMDIST(number_s, trials, probability_s, cumulative)
number_s = cell location of x
trials = how many times you are performing experiment
probability_s = probability of success
cumulative = “false” for pmf; “true” for cdf

13. Review of Finite Random Variables
Finite R.V takes on a set of discrete values (you can list all of the numerical values)
Probability Mass Function (p.m.f) describes the probability distribution
fx(x) where fx(x) = P( X = x )
graph is a histogram
sum of the heights of the rectangles must equal one
Cumulative Distribution Function (c.d.f)
FX(x) = P(X  x)
graph is a step function
minimum is 0 and the maximum is 1

14. Review of Finite Random Variables
Binomial Random Variable is a random variable that stands for the number of successes in n Bernoulli Trials
A Bernoulli Trial has only 2 possible outcomes: success and failure
Binomial Setting:
You have n repeated trials of an experiment
On a single trial, only two possible outcomes
The probability of success is the same from trial to trial
The outcome of each trial is independent
Expected Value is n(p), where p is the probability of success and n is the number of trials of the experiment

15. Continuous Random Variable
Continuous random variables take on values in an interval; you cannot list all the possible values
Examples: 1. Let X be a randomly selected number between 0 and 1 2. Let R be a future value of a weekly ratio of closing prices for IBM stock 3. Let W be the exact weight of a randomly selected student
You can only calculate probabilities associated with interval values of X. You cannot calculate P(X=x); however we can still look at its c.d.f, FX(x).

16. Probability Density Function (p.d.f)
When we looked at finite random variables, we created a p.m.f graph (histogram)
Our graph had rectangles with a certain width
This width was the distance between two values of the random variable
When we start to make our width smaller and smaller, we begin to see a curve

17. Probability Density Function (p.d.f)
When we look at continuous random variables, we are looking at random variables that take on every value in a given interval
The width of our rectangles are now infinitesimally small
When we look at this histogram, we are approximating our p.d.f
When we graph all of the values of the continuous r.v, our p.d.f graph looks like a curver
This graph is called the graph of the Probability Density Function (p.d.f)
Probability Density Function is represented by fX(x)

18. Probability Density Function
Below is an example of a p.d.f graph
Note: The notation for a pmf and a pdf are the same (fX(x)) – you will need to be careful about the interpretation of the function A a b fX

19. Probability Density Function (p.d.f)
For the graph of the p.m.f, the values along the y-axis (the probabilities) summed up to 1
The same holds true for the p.d.f graph
The area under the curve adds up to 1 (because the area under the graph represents the total probability)
Note: There is no one type of curve that you are looking for – there are different types of continuous random variables so the graphs of the pdf will look different

20. How to tell if the graph is a p.d.f?
We use the word “curve” but the graph could be a straight line
We could also have a histogram that is approximating the p.d.f.
If the area under the graph is 1, then the graph represents a p.d.f.
If the graph is a histogram, how can you tell what function it represents?
In the finite case, the sum of the heights of the rectangles add up to 1
In the continuous case, the sum of the heights of the rectangles do add to 1 but the areas of the rectangles do sum up to 1

21. Probability Density Function (p.d.f)
For any continuous random variable, X, P(X=a)=0 for every number a.
Instead of considering what the probability of X is at a single value, we look for the probability that X assumes a value in an interval
P(a  X  b) is the probability that X assumes a value in [a,b]

22. Probability Density Function (p.d.f)
To find P(a  X  b), we need to look at the portion of the graph that corresponds to this interval. A a b fX

23. Finding Values of the pdf
To find the probabilities associated with the pdf, you can calculate them in two ways
You can look at the area under the curve associated with the inteval in question
Do this when you are given the pdf function
For example, look at #6 on the random variable worksheet
You can use the cdf
Do this when you are given the cdf function

24. Calculating P(a  X  b) from a p.d.f

25. Probability Density Function

26. Cumulative Distribution Function
The same probability information is often given in a different form, called the cumulative distribution function, (c.d.f), FX
FX(x)=P(Xx)
0  FX(x)  1, for all x
NOTE: Regardless of whether the random variable is finite or continuous, the cdf, FX, has the same interpretation
I.e., FX(x)=P(Xx)

27. Cumulative Distribution Function
For the finite case, our c.d.f graph was a step function
For the continuous case, our c.d.f. graph will be a continuous graph
Note: The minimum is still 0 and the maximum is still 1

28. Cumulative Distribution Function
Now, depending on the type of continuous random variable, the graph of the cdf will look different
Below is an example of a graph of a cdf for a continuous random variable

29. Review of Continuous Random Variables
Continuous R.V. takes on any value in a given interval; you cannot list all of the values
Probability Density Function (p.d.f.) describes the distribution of the probabilities
fX(x) where fX(x) does not equal P( X = x )
fX(x) simply represents the height of the curve at a given value of the random variable
We can only calculate the probabilities of intervals
to calcuate P(a  X  b) -- use the graph of the p.d.f and find the corresponding area under the curve OR calculate FX(b) - FX(a) if given the c.d.f
P(X=a)=0 for every number a

30. Review of Continuous Random Variables
Cumulative Distribution Function (c.d.f)
FX(x) = P(X  x)
graph is an increasing function with minimum at 0 and maximum at 1
Note! At every new value of a R.V. (finite or continuous), a c.d.f adds on the associated probability of the new value of the R.V
for finite R.V., it looks like a step funciton since there are only a finite amount of number
for continuous R.V., it a continuous increasing function
both graphs have minimum at 0 and maximum at 1

31. Special Types of Continuous R.V.
Exponential random variables usually describe the waiting time between consecutive events.
In general, the p.d.f and c.d.f for an exponential random variable X is given as follows:
 (pronounced alpha) is a Greek letter – it represents a number in the formula
Remember! P(a<X<b) = FX(b) - FX(a) AND P(X=a)=0 because an exponential random variable is a continuous random variable

32. Continuous R.V. with exponential distribution

33. Special Types of Continuous R.V.
If the probability that X assumes a value is the same for all equal subintervals of an interval [0,u], then we have a uniform random variable, where u is the interval length
X is equally likely (probabilities are equal) to assume any value in [0,u]
If X is uniform on the interval [0,u], then we have the following formulas:

34. Continuous R.V. with uniform distribution

35. Expected Value
From Project 1, Expected Value of a Finite Random Variable is
This can now be written as
This is called the mean of X
It is denoted by X
For a Binomial Random Variable, E(X)=n*p, where n is the the number of independent trials and p is the probability of success

36. Expected Value
If X is continuous, you cannot sum over all the values of X, since P(X=x)=0 for all x
In general, if X is a continuous random variable with a UNIFORM distribution on [0,u], then
Any EXPONENTIAL random variable X, with parameter , has

37. FOCUS ON THE PROJECT
GOAL: To price a European call option on the option’s starting date
For our project, we are using several Random Variables
C, the closing price per share
R, the ratio of closing prices
Rm is the mean of the ratios of closing prices
Rnorm is the continuous r.v. of normalized ratios
Rnorm = R – (Rm-Rrf)

38. Focus on the project
Use ratios to estimate the basic volatility of stock
Normalize ratios first (IMPORTANT!)
Why? Want to compare how the stock is doing to what the money is doing in bank (at risk-free rate)
From each ratio, you are going to subtract out the growth rate of the stock but leave the trend (thus making the growth rate 0)
To each ratio, you are going to add in the carrying cost – the growth at the risk-free rate (this is from assumption number 4)

39. Focus on the project
How to normalize? Adjust observed ratios so average is same as risk-free weekly ratio
I.e., reduce observed ratios by the difference Rm-Rrf
Now, recall that Rnorm = R – (Rm-Rrf)
Note, the average value of normalized ratios is Rrf= Rm-(Rm-Rrf)

40. What should you do?
Since you have all of your ratios (found for homework #6), you should normalize each of them.
I.e, For each ratio that you have, you will need to subtract Rm-Rrf from each ratio R.
This gives you a Rnorm for each ratio
To do this:
You will need to find the mean of the ratios
Use the weekly ratio at the risk-free rate