Download - 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