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Finite 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)
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 )
1
all x
( ) 1. Thus, 0 ( ) 1 for any value of and
( ) 1
n
i Xi
X
P X x f x x
f x
Probability Mass Function
This is a p.m.f which is a histogram representing the probabilities
The bars are centered above the values of the random variable
The heights of the bars are equal to the corresponding probabilities (when the width of your rectangles is 1) 0
0.1
0.2
0.3
0.4
0.5
0 1 2
P(X=x)
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.
Cumulative Distribution Function
Cumulative Distrib ution Function
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
x
F X (x )
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 outcomes3. The probability of success is the same from trial to trial4. The outcome of each trial is independent
Expected Value of a Binomial R.V is represented by E(X)=n*p
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– BINOMDIST(x, n, p, false) = P(X=x)– BINOMDIST(x, n, p, true) = P(X ≤ x)
Expected Value
This is average value of X (what happens on average in infinitely many repeated trials of the underlying experiment
– 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
x
X xfxXEall
)()(
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(X=x); however we can still look at its c.d.f, FX(x).
Probability Density Function (p.d.f)
Represented by fx(x)– fx(x) is the height of the function fx(x) at an input of x
– This function does not give probabilities
For any continuous random variable, X, P(X=a)=0 for every number a.
Look at probabilities associated with X taking on an interval of values– P(a ≤ X ≤ b)
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.
How can we relate this to integration?
Aa b
fX
Cumulative Distribution Function
CDF --– 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)
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
Cumulative Distribution Function
0.00.20.40.60.81.01.2
-1 0 1 2 3t
F T(t )
Fundamental Theorem of Calculus (FTC)
Given that – Differentiate both sides and what happens?
Well, from the previous slide we can see that
– If we differentiate both sides, we get that
What does this say? How can we verify this claim?
dxxgxG )()(
dxxfxF XX )()(
)()(' xfxF XX
Example 7 from Course Files
Define the following function:
– What are the possible values of X?– Set up an integral that would give you the following
probabilities: P(X < 0.5) P(X > 0.6) P(0.1 ≤ X ≤ 0.9) P(0.1 ≤ X ≤ 5)
– Verify that the function is a density function – What is E(X)?
elsewhere 0
10 if 155.37305.7)(
234 xxxxxxf X
Expected Value
For a finite random variable, we summed over all possible values of x
For a continuous random variable, we want to integrate over all possible values of x
This implies that
dxxfxXE XX )()(
Example 8 from the Course Files
Let T be the amount of time between consecutive computer crashes and has the following p.d.f. and c.d.f.
– What type of r.v. is T?– Calculate P(1 < T < 5) in
two different ways.– What is E(X)?
0 tif 1
0 if 0)(
0 tif 8.16
1
0 if 0
)(
8.16
8.16
tT
tT
e
ttF
e
t
tf
Exponential Distribution
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:
Any EXPONENTIAL random variable X, with parameter , has
How can we verify this?
xe
xxF xX 0if1
0if0)( /
xe
xxf xX 0if
10if0
)( /
( )E X
Continuous R.V. with exponential distribution
Probability Density Function
0.00.10.20.30.40.50.6
-3 0 3 6 9 12 15x
f X (x )
Cumulative Distribution Function
0.00.20.40.60.81.01.2
-3 0 3 6 9 12 15x
F X (x )
• How can we verify that the graph on the left is the graph of a p.d.f.?
Uniform Distribution
If the probability that X assumes a value is the same for all equal subintervals of an interval [0,u], then we have a continuous uniform random variable
X is equally likely to assume any value in [0,u] If X is uniform on the interval [0,u], then we have the
following formulas:
xu
uxu
ux
xf X
if0
0if1
if0
)(
xu
uxu
x
ux
xFX
if1
0if
if0
)(
Continuous R.V. with uniform distribution
Probbility Density Function
0.0000
0.0004
0.0008
0.0012
0.0016
-100 0 100 200 300 400 500 600 700 800
x
f X (x )
Cumulative Distribution Function
0.0
0.2
0.4
0.6
0.8
1.0
-100 0 100 200 300 400 500 600 700 800
x
F X (x )
• In general, if X is a continuous random variable with a UNIFORM distribution on [0,u], then
( )2
uE X
Focus on the Project
Look at the file Auction Focus.xls in the course files– This file contains 22 prior leases– Looking at each prior lease, we see that if each company bid
their signal, every company that won the auction would have lost money
– We want to devise a new bidding strategy using this data
Use data to simulate thousands of similar auctions
Identify Random Variables
We need random variables– Let V be the continuous random variable that gives the fair
profit value, in millions of dollars, for an oil lease similar to the 22 tracts
Look through Auction Focus.xls to see the statistics for the sample
– Each signal is an observation of the continuous random variable, SV where v is the actual fair value of the tract
It is assumed that E(SV) = v for every lease
– RV gives the error in a company’s signal Given by the signal minus the actual fair profit value of the lease E(RV) = 0 for every value of v
What should you do?
From slide 65 in MBD 2 Proj 2.ppt –1. Start an Excel file which incorporates the historical data on
the lease values and your team’s particular set of signals
2. Use these to compute the complete sample of signal errors, and then analyze this sample. Specifically, you should compute the maximum, minimum, and sample mean of the errors. You should also plot a histogram that approximates the actual p.d.f, fR of R– Go to slide 50 to see information about relative frequencies