Probability Distributions Random Variables: Finite and Continuous Distribution Functions Expected value

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Probability DistributionsRandom Variables
Finite and ContinuousDistribution
FunctionsExpected value

• April 3 10, 2003

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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 X0,1,2, 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
  X1,2,3,4,)
• Continuous If the possible values form an entire
  interval of numbers (ex any positive number)

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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)

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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 )

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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 0 1 2
P(Xx) or fX .25 .50 .25
From this table, we have the ordered pairs
(0,.25),(1,.5),(2,.25)
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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(Xx) 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

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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.

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Cumulative Distribution Function
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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

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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

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Binomial Random Variable

• Let X stand for the number of successes in n
  Bernoulli Trials where X is called a Binomial
  Random Variable
• Binomial Setting1. 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)np

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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

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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

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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

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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 12. Let R be a future
  value of a weekly ratio of closing prices for
  IBM stock3. 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(Xx) however we can still look at its c.d.f,
  FX(x).

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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

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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)

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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

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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

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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

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Probability Density Function (p.d.f)

• For any continuous random variable, X, P(Xa)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

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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.

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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

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Calculating P(a ? X ? b) from a p.d.f
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Probability Density Function
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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)

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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

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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

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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(Xa)0 for every number a

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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

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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(altXltb) FX(b) - FX(a) AND P(Xa)0
  because an exponential random variable is a
  continuous random variable

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Continuous R.V. with exponential distribution
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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

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Continuous R.V. with uniform distribution
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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)np, where n
  is the the number of independent trials and p is
  the probability of success

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Expected Value

• If X is continuous, you cannot sum over all the
  values of X, since P(Xx)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

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FOCUS ON THE PROJECT

• GOAL To price a European call option on the
  options 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)

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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)

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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)

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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