random variable - transformations

7/31/2019 Random Variable - Transformations

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Probability


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Probability1. Probability

2. Conditional Probability, Bayes Theorem3. Independent Trials, Bernoullis Distribution

. 1. CDF, PDF, Conditional CDF

2. Functions of Random Variables

3. Characteristic Function4. Expectation, Moments, Central Moments

5. Markov and Chebyshev Inequality


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Probability

Two or More Random Variables

Joint CDF

Correlation, Covariance

. . Two Functions of Two R.V.s

Gaussian Random Variables

Covariance Matrix - Eigen Decomposition

Quadratic Form


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Functions of one Random Variable

Case 1: g(.) is monotonically increasing


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Example

Suppose X is a Gaussian random variable

with mean, , and variance, . A new randomvariable is formed according to Y = aX + b,

.


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Example

Suppose X is a Gaussian random variable

with mean, , and variance, . A new randomvariable is formed according to Y = aX + b,

.


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Example

Suppose a phase angle is uniformly

distributed over ( /2, /2), and thetransformation is Y = sin()


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Characteristic Functions

Similarity to Fourier Transform

(-) Not associated with any physical frequency

Computational convenience e.g., convolving PDFs


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Example An exponential random variable has a PDF

given by fX(x) = exp(x)u(x). Find itscharacteristic function.


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Example An exponential random variable has a PDF

given by fX(x) = exp(x)u(x). Find itscharacteristic function.


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Example Another random variable Y has a PDF given by

fY(y) = a exp(ay)u(y). Find its characteristicfunction.


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Example Another random variable Y has a PDF given by

fY(y) = a exp(ay)u(y). Find its characteristicfunction.

,


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Tail Probabilities Compute the probability that a random

variable exceeds a threshold, Pr(X > xo)

Compute the probability that a random

, x

o

Computing from CDF or PDF may be

cumbersome

Can we obtain a bound, if not the actualprobabilities ?


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Markov Inequality

XgE )]([

X is a random variable. If g(X) is a non negative

function of X g(X) 0 for all X

k


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Markov Inequality Suppose that X is a nonnegative random

variable


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Example Suppose the average life span of a person is 75

years. What is the probability of a humanliving to be 110 years ?


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Chebyshevs Inequality Suppose that X is a random variable with

mean X and variance X2


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Example Suppose the average life span of a person is 75

years. The human lifespan has a SD of 5 yearsWhat is the probability of a human living to be


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Conditional CDF In a company, resistors are required to have

a resistance R of 50 2 . Owing toimprecision in the manufacturing process, the

shown below.


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Conditional CDF The quality department then screens and

discards resistors outside the required range.What is the CDF after this process ?


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Soln


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Pblm The survival of a motorist stranded in a

snowstorm depends on which of the threedirections the motorist chooses to walk. The

travel, the second leads to safety after three

hours of travel, but the third will circle back to

the original spot after two hours. Determine

the average time to safety if the motorist is

equally likely to choose any one of the roads.


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Moments and Expectation


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Soln.


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Joint CDF A fair coin is tossed three times. Let X be a

random variable that takes the value 0 if thefirst toss is a tail and the value 1 if the first

. ,

that defines the total number of heads in the

three tosses.

a. Determine the joint PMF of X and Y.b. Are X and Y independent?


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Joint CDF A and B decide to meet at a certain place

between 9 a.m. and 10 a.m. Both of themwont wait for more than 10 minutes. If all

times are independent, what is the probability

that they would meet ?


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Joint CDF


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Joint CDF P(they will meet) = P(|X-Y| 10)


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Correlation and Covariance Covariance

Correlation


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Correlation Suppose X is a normally-distributed random

variable with zero mean and Y = X2. Computetheir correlation coefficient.


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One Function of Two R.V.s


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Z = X + Y


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Z = X + Y

Take the inverse transform


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One Function of Two R.V.s Suppose X and Y are independent, zero-mean,

unit variance Gaussian random variables.Find the PDF of Z = X2 + Y2


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One Function of Two R.V.s Suppose X and Y are independent, zero-mean,

unit variance Gaussian random variables.Find the PDF of Z = X2 + Y2

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