Probability, Stochastic Processes and Statistics (19I401) - Course Syllabus, Assignments of Probability and Stochastic Processes

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19I401 -

PROBABILITY, STOCHASTIC PROCESSES AND STATISTI

CS

19I401 -

PROBABILITY, STOCHASTIC PROCESSES AND STATISTICS

Chapter 1: PROBABILITY AND DISCRETE RANDOM VARIABLES Chapter 2: CONTINUOUS RANDOM VARIABLES Chapter 3: PAIRS OF RANDOM VARIABLES Chapter 4: STOCHASTIC PROCESSES Chapter 5: STATISTICAL INFERENCE

Week -

Discrete random variables :

• (^) Probability mass function
• (^) Cumulative distribution functions
• (^) Expectations

Text Books:

T1. Roy D Yates and David J Goodman, “Probability and Stochastic Processes - A

friendly Introduction for Electrical and Computer Engineers”, Wiley India, New

Delhi, 2014.

T2. Ronald E. Walpole, Raymond H Myers, Sharon L Myers and Keying Ye,

“Probability and Statistics for Engineers and Scientists”, Pearsons, New Delhi, 2016

Random Variable - Introduction Motivation: In most statistical problems we are concerned with one number or a few numbers that are associated with the outcomes of experiments. When inspecting a manufactured product we may be interested only in the number of defectives; in the analysis of a road test we may be interested only in the average speed and the average fuel consumption; and in the study of the performance of a miniature rechargeable battery we may be interested only in its power and life length. All these numbers are associated with situations involving an element of chance—in other words, they are values of random variables.

In a random experiment, the sample space ‘S’ consists of all outcomes (results) of that experiment. When the elements of the sample space are non numeric, they can be quantified by assigning a real number to every event of the sample space. A real number X associated with the outcome of a random experiment. A random variable is a real valued function defined on the sample space Random variable is also known as stochastic variable or variable. Thus to each outcome `S’ , there corresponds a real number X(s). Random variables are denoted by capital letters X,Y, and so on, to distinguish them from their possible values given in lowercase x, y. Random Variable - Introduction

A probability model always begins with an experiment. Each random variable is related directly to this experiment. There are three types of relationships.

1. The random variable is the observation.
2. The random variable is a function of the observation.
3. The random variable is a function of another random variable. Random Variable A random variable consists of an experiment with a probability measure P [·] defined on a sample space S and a function that assigns a real number to each outcome in the sample space of the experiment. Here are some more random variables:

• A , the number of students asleep in the next probability lecture;
• C , the number of phone calls you answer in the next hour;
• M , the number of minutes you wait until you next answer the phone. Random variables A and C are discrete random variables. The possible values of these random variables form a countable set. The underlying experiments have sample spaces that are discrete. The random variable M can be any nonnegative real number. It is a continuous random variable. Its experiment has a continuous sample space

Random Variable – Definition & Example

Random Variable – Classification

Random variables are usually classified according to the number of values they can assume. Random variable can be classified as

1. discrete random variables, which can take on only a finite number
2. a countable infinity of values; called continuous random variables Examples:

• (^) Life time of a machine in years – Discrete
• (^) Expected mark in the upcoming midsemester exam – Continuous