Download Probability and Stochastic Processes

Unit-II
Random variables and operations on One
random variable
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Course Objective
• Characterize probability models by employing counting methods and basic
probability mass function and probability density function canonical
models for discrete and continuous random variables. It give different
operations on single random variables like expectation.
• evaluation of first and second moments and cumulative distribution
functions for both discrete and continuous random variables.
Chapter 0
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2.1 Definition of a Random Variable, Types of Random Variables.
2.2 Conditions for a Function to be a Random Variable
2.3 Distribution and Density functions, and their Properties2.4 Binomial, Poisson, Uniform ,Gaussian,
2.5 Conditional Distribution, Conditional Density, Properties.
Introduction, Expected Value of a Random Variable,
2.7 Moments about the Origin, Central Moments,
2.8 Variance and Skew,
2.9 Characteristic Function, Moment Generating Function,
2.10 Transformations of a Random Variable: Monotonic Transformations
for a Continuous Random Variable,
2.11Non monotonic Transformations of Continuous Random Variable,
2.12Transformation of a Discrete Random Variable.
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2.1Definition of a Random Variable, Types of
Random Variables.
• A random variable X can be considered to be a function that maps all
elements of the sample space into points on the real line or some parts
thereof.
• We define a real random variable as a real function of the elements of a
sample space S.
• Definition of Random Variable A random variable is a function from a
sample space S into the real numbers.
• Types of random variables 1.Discrete random variable 2. Continuous
random variable 3.Mixed random variable.
1.Discrete random variables: where the possible events are countable. For
example, the roll of a dice, or the outcome of a horse race, or whether the
firm will default or not.
2.Continuous random variables: where the possible events are not
countable. For example, the number of white hair on my head, or how
much dividend INFOSYSTCH will announce next year, or the price of
Citibank stock.
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2.2Conditions for a Function to be a Random
Variable
A random variable may be almost any function we wish.
We shall ,however ,require that it not be multi valued. That is ,every
point in S Must correspond to only one value of the random variable.
The second condition we require is that the probabilities of the events
{X=infinite} and {X=-infinite} be 0.
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2.3 Distribution and Density functions, and their
Properties• What is a probability distribution?
• For a discrete RV, the probability distribution (PD) is a table of all the
events and their related probabilities.
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• A probability distribution will contain all the outcomes and
their related probabilities, and the probabilities will sum to
1.
• How to read a probability distribution?
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• What is a cumulative probability distribution (CD)?
• A table of the probabilities cumulated over the events.
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Distribution properties
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Properties of the PDF
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2.4 Binomial Distribution
“n independent coin flips” p = Pr(success)
N = # of successes
 n k
nk
Pr  N  k     p 1  p  , k  0,1,..., n
k
E  N   np
Var  N   np 1  p 
1 p
Cv 
np
2
N
M    1  p  pe
*

 n
Chapter 0
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2.4 Poisson Distribution
“Occurrence of rare events”  = average rate of occurrence
per period;
N = # of events in an arbitrary period
 k e 
Pr  N  k  
, k  0,1, 2,...
k!
EN  
Var  N   
CvN2  1 
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2.4 Uniform Distribution
X is equally likely to fall anywhere within interval (a,b)
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fX  x 
, a xb
ba
EX  
Var  X 
ab
2
b  a


2
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b  a


2
3b  a 
2
Cv X2
a
b
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2.4Normal Distribution
X follows a “bell-shaped” density function
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 x
2
fX  x 
e 
,   x  
 2
2
2
EX   
Var  X    2
From the central limit theorem, the distribution of the sum of
independent and identically distributed random variables
approaches a normal distribution as the number of summed
random variables goes to infinity.
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2.5 Conditional Distribution, Conditional Density, Properties.
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2.6 Introduction, Expected Value of a Random Variable
• In probability theory , the expected value of a random variable is
intuitively the long-run average value of repetitions of the experiment
it represents. For example, the expected value of a dice roll is 3.5
because, roughly speaking, the average of an extremely large number
of dice rolls is practically always nearly equal to 3.5.
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2.7 Moments about the Origin, Central Moments,
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2.7 Moments about the Origin, Central Moments
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2.8Variance and Skew,
• In probability theory and statistics, variance measures how far a set of
numbers is spread out. A variance of zero indicates that all the values
are identical. Variance is always non-negative: a small variance
indicates that the data points tend to be very close to
the mean (expected value) and hence to each other, while a high
variance indicates that the data points are very spread out around the
mean and from each other.
• An equivalent measure is the square root of the variance, called
the standard deviation .
• In probability theory and statistics , skewness is a measure of the
asymmetry of the probability distribution of a real -valued random
variable about its mean. The skewness value can be positive or
negative, or even undefined.
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2. 9 Characteristic Function,
Moment Generating Function,
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2.10Transformations of a Random Variable: Monotonic
Transformations for a Continuous Random Variable,
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2.11Non monotonic Transformations of Continuous Random
Variable,
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2.12Transformation of a Discrete Random Variable.
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