Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Problem Set 8 Solutions
1. Let At (respectively, Bt ) be a Bernoulli random variable that is equal to
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
LECTURE 23
Readings: Section 7.4, 7.5 Lecture outline
Proof of the central limit theorem Approximating binomial distributions
CLT Review
standard normal (zero mean, unit variance) : i.i.d. finite variance variance
CLT: For every
Normal approximation:
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
LECTURE 22
Readings: Section 7.4 Lecture outline
The Central Limit Theorem:
Introduction Formulation and interpretation Pollsters problem Usefulness
Introduction
i.i.d. finite variance
Look at three variants of their sum: variance converges in probab
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
LECTURE 15 Readings: Sections 7.17.3
Lecture outline Limit theorems:
Chebyshev inequality Convergence in probability
Motivation
i.i.d., (sample mean) What happens as Why bother? A tool: Chebyshevs inequality. Convergence in probability. Convergence of .
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
LECTURE 4
Convergence and Asymptotic
Equipartition Property
Last time: Fanos Inequality Stochastic Processes Entropy Rate Hiden Markov Process Lecture outline Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers
Asymptotic Equipa
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Tutorial 9 April 2021, 2006 1. SignaltoNoise Ratio:If random variable X has mean = 0 and standard de
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Tutorial 8 April 1314, 2006 1. b (X, Y ) = . b2 + 2c2 YLinear (X ) = a + c + bX 2. (a) See solutions to
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Tutorial 08 April 1314, 2006 1. Suppose X is a unit normal random variable. Dene a new random variable
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Solutions for Recitation 22 Central Limit Theorem May 16, 2006
1. See solution in text, page 390. 2. Se
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Recitation 22 Central Limit Theorem May 16, 2006
1. (Example 7.8) We load on a plane 100 packages whose
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Recitation 15 April 20, 2006 1. See textbook pg. 399 2. (a) N = 200, 000. (b) N = 100, 000. 3. Let us x
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Recitation 15 April 20, 2006 1. Let X and Y b e random variables, and let a and b b e scalars; X takes
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Solutions for Problem Set 12: Topic: Central Limit Theorem Due: No due date
1. It is not easy to calcul
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Problem Set 12:
Topic: Central Limit Theorem
Due: No due date
1. The weight of a Pernotti Parabolic
Probabilistic Systems Analysis and Applied Probability
MATH 6.041 / 6.

Spring 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Problem Set 8 Topics: Covariance, Estimation, Limit Theorems Due: April 26, 2006 1. Consider n independ