The Markov Inequality
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Suppose x is a non-negative random variable
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Markov inequality upper bounds the probability that x is large
P (x a)
E [x]
a
(1)
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To use it only information that is needed is the mean of the
r.v.
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No need for the P.M.F/P.D.F of
Lecture 1
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Course Information: PRP
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Class Hours : Wed: 11:30 to 12:45, Fri: 10:00 to 11:15
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Office Hours: Thursday: 1:00 to 2:00 P.M
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TAs: TBD
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Website: www.iiitd.edu.in/~praveshb/teaching.html
Text
Lecture 3: Conditional Probability and Stochastic
Independence
IIIT Delhi
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August 17, 2016
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Conditional Probability
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Probability given an event (B) has occurred
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Revision of belief after knowing about an event
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Definition:
P(A
Random Processes
IIIT Delhi
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Bernoulli Process
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A sequence of independent Bernoulli trials
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At each trial i
1. P (success) = P (Xi = 1) = p
2. P (failure) = P (Xi = 0) = 1 p
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Examples:
1. Sequence of lottery
Lecture 4: Random Variables, Discrete PMFs,
Expectation and Variance
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August 26, 2016
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Random Variable
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R.Vs is way of assigning numerical values to outcomes
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Function from sample space to Real line R.
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Can have mul
Conditional P.M.F, Expectations and Joint
Distribution
IIIT Delhi
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August 31, 2016
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Conditional P.M.F
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Random variable X has P.M.F pX (x)
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A is an event
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Conditional P.M.F - pX |A (x) = P (X = x | A)
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All properties of P.M.Fs ho
Continuous Random Variables: Bayes Rule,
Functions of Random Variable(s)
IIIT Delhi
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September 30, 2016
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Continuous R.V: Probability Density Function (PDF)
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FX ,Y (x, y ) = P (X x, Y y )
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fX ,Y (x, y ) =
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R
Marginals from join
Continuous Random Variables: PDF,CDF
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September 14, 2016
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Continuous R.V: Probability Density Function (PDF)
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A continuous R.V is described by a PDF fX
Z
P (a x b) =
b
fX (x) dx
a
Z
fX (x) = 1
P (x x x + x) fX (x) x
Z
P
Principles of Digital Communication System
Assignment 2
Deadline: Nov 2, 2016
1. Consider the three waveforms fn(t) shown in figure.
(a) Show that these waveforms are orthonormal.
(b) Express the waveform x(t) as a weighted linear combination of fn(t), n
Lecture 2: Counting
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August 10, 2016
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Basic Principles of Counting
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r stages
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ni choices at stage i
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Total number of choices is: n1 n2 . . . nr
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Example: How many license plates with 3 letters and 4 digits?
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Example:
Probability and Random Processes
ECE 501
Assignment 1
1. Letters in the Morse code are formed by a succession of dashes and dots with
repetitions permitted? How many letters is it possible to form with ten symbols
or less?
2. Each domino piece is marked b