Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.

3.2

DISCRETE RANDOM VARIABLE

A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probability distribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6

The probability distribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1

QMT200

3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.

= ( ) Where: = ∑[ ( )]

( )=

= (

) − [ ( )]

(

)=

[

( )] = ( )

Example 1 An experiment consists of tossing two coins simultaneously. Write down the sample space. If X is the number of tails observed, determine the probability distribution of X.

QMT200

Example 2 The following table shows the probability of the number of long-distance telephone calls made in a month by residents of a sample of urban households. X 0 1 2 3 4 5 6 7 8 9 10 P(X=x) 0.02 0.05 0.08 0.11 0.14 0.22 0.28 0.04 0.03 0.02 0.01

a) Find the mean number of calls per household. b) Calculate the variance c) Find (1 < ≤ 6)

QMT200

3.2.2 MEAN AND VARIANCE OF LINEAR COMBINATIONS OF RANDOM VARIABLES Rules of Mean For any constant a and b, a) b) c) d) ( )= ( )= ( ) ( ± )= ( )± ( ) ( ± )= ( )± ( ) Rules of Variance For...

...PROBABILITY and MENDELIAN GENETICS LAB
Hypothesis: If we toss the coin(s) for many times, then we will have more chances to reach the prediction that we expect based on the principle of probability.
Results:
As for part 1: probability of the occurrence of a single event, the deviation of heads and tails of 20 tosses is zero, which means that the possibility of heads and tails is ten to ten, which means equally chances. The deviation of heads and tails of 30 tosses is 4, which means that the occurrence of head is 19 and the occurrence of tail is 11. The deviation of heads and tails of 50 tosses is 3, which means that the occurrence of head is 28 and the occurrence of tail is 22. Compare the second and third observation, we can find that the deviation decrease one. It is corresponding to the hypothesis. The more times we use to toss the coin, the more opportunities we will get to reach the prediction based on the principle of probability.
As for part 2: probability of independent events occurring simultaneously, the observation of Heads-Heads is 11, which is 27.5% of the total experiment. And the deviation is 1. The observation of Heads-Tails or Tail-Heads is 16, which is 40% of the total. And the deviation is 4. The observation of Tails- Tails is 13, which is 32.5% of the total number. And the deviation is 3.
Discussion:
In this lab, we’d learn about the likelihood that a particular event will...

...Learning Programmes Division
Second Semester 2010-2011
Course Handout
Course Number
Course Title
: AAOC ZC111
: Probability and Statistics
Course E-mail address : aaoczc111@dlpd.bits-pilani.ac.in
Course Description
Probability spaces; conditional probability and independence; random variables and probability
distributions; marginal and conditional distributions; independent random variables, mathematical
exceptions, mean and variance, Binomial Poisson and normal distribution; sum of independent random
variables; law of large numbers; central limit theorem; sampling distributions; tests for mean using normal
and student’s distributions; tests of hypotheses; correlation and linear regression.
Scope and Objectives
At the end of the course, the student should be able to understand probabilistic & deterministic models and
statistical inference and apply these concepts to solve a variety of problems.
Prescribed Text Book
T1
Johnson Richard A. & C.B. Gupta, Miller & Freund’s Probability and Statistics for Engineers, PHI,
7th Ed., 2005.
Reference Books
R1.
Paul L. Meyer, Introductory Probability and Statistical Appl., Second Edition. Addison-Wesley, 1970.
R2. M.S. Radhakrishnan, Probability & Statistics, DLPD Notes Note: Softcopy of this Supplementary notes will
be available for download from BITS DLP website.
R3. Mendenhall Beaver Beaver, Introduction to...

...of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean also of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance - The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation - The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution - is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial...

...TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Chapter 1: Discrete and Continuous Probability Distributions
Section 1: Probability
Contents: 1.1 1.2 1.3 1.4 1.5 Some basics of probability theory Axioms, Interpretations, and Properties of Probability Counting Techniques and Probability Conditional Probability Independence
TEM1116
1
TEM1116Probability and Statistics
Tri1 2013/14
Chapter 1
1.1
Basics of Probability Theory
Probability refers to the study of randomness and uncertainty. The word “probability” as used in “probability of an event” is a numerical measure of the chance for the occurrence of an event. Experiment: a repeatable procedure with a well-defined set of possible outcomes. (Devore: Any action or process whose outcome is subject to uncertainty.) Sample Space and Events Sample space of an experiment is the set of all possible outcomes. An event is a set of outcomes (it is a subset of the sample space). Example: Consider an experiment of rolling a 6-sided die.
Sample Space, S :
{1, 2, 3, 4, 5, 6}
S
Events, Ek: E1: composite number is rolled. → Equivalently, {4, 6}. E2: number less than four is rolled. → Equivalently, {1, 2, 3}.
E1
E2
Example 1.1 : An experiment consists of tossing three coins. Find the sample space if (i) We are...

...Contents | Page |
Part 1 | 2 |
Part 2 | 5 |
Part 3 | 6 |
Part 4 | 8 |
Part 5 | 10 |
Further Exploration | 13 |
Conclusion | 14 |
Reflection | 15 |
Part 1
a) Introduction
The word Probability derives from probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which, in contrast, is used as a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.
A short history of Probability Theory............
The branch of mathematics known as probability theory was inspired by gambling problems. The earliest work was performed by Girolamo Cardano (1501-1576) an Italian mathematician, physician, and gambler. In his manual Liber de Ludo Aleae, Cardano discusses many of the basic concepts of probability complete with a systematic analysis of gambling problems. Unfortunately, Cardano's work had little effect on the development
of probability because his manual, which did not appeared in print until 1663, received little attention.
In 1654, another gambler named Chevalier de Méré created a dice proposition which he believed would make money. He would bet even money that he could roll at least one 12 in 24 rolls of two dice. However, when the Chevalier began losing money, he...

...Probability Theory and Game of Chance
Jingjing Xu
April 24, 2012
I. INTRODUCTION
Probability theory is the mathematical foundation of statistics, and it can be applied to many areas requiring large data analysis. Curiously, that the study on probability theory has its root in parlor games and gambling. In 17th century, dice gambling was a very common entertainment among the upper class. An Italian mathematician and gambler Gerolamo Cardano founded the concept of probability by studying the rules of rolling dice: since a die is a cube with each of its six faces showing a different number from 1 to 6, when it is rolled, the probability of seeing each number is equal. Therefore, some of the gamblers began to wonder, that taking a pair of dice and rolling them a couple of times, which has the larger probability of seeing a sum of 9 or seeing a sum of 10? What about seeing double sixes? In a correspondence between Blaise Pascal and Pierre Fermat, the problems were resolved, and this triggered the first theorem in the modern theory of probability.
II. BASIC DEFINITIONS
Definition 1
In probability theory, the sample space, often denoted Ω, of an experiment is the set of all possible outcomes....

...pairs of shoes. Five shoes are drawn at random. What is the probability that at least one pair of shoes is obtained? 2. At a camera factory, an inspector checks 20 cameras and ﬁnds that three of them need adjustment before they can be shipped. Another employee carelessly mixes the cameras up so that no one knows which is which. Thus, the inspector must recheck the cameras one at a time until he locates all the bad ones. (a) What is the probability that no more than 17 cameras need to be rechecked? (b) What is the probability that exactly 17 must be rechecked? 3. We consider permutations of the string ”ABACADAFAG”. How many permutations are there? How many of them don’t have any A next to other A? How many of them have at least two A’s next to each other? 4. A monkey is typing random numerical strings of length 7 using the digits 1 through 9 (not 0). Call the digits 1, 2, and 3 ”lows”, call the digits 4, 5, and 6 ”mids” and digits 7, 8 and 9 ”highs”. (a) How many diﬀerent strings can he type? (b) How many of these strings have no mids? (c) How many of these strings have only one high in them? For example, the string 1111199 has two highs in it. (d) What’s the probability that a string starts with a low and ends with a high? (e) What’s the probability that a string starts with a low or ends with a high? (f) What’s the probability that a string doesn’t have at least one of the digits 1 through...

...Important Discrete
Probability Distributions
5-1
Chapter Goals
After completing this chapter, you should be able
to:
Interpret the mean and standard deviation for a
discrete probability distribution
Explain covariance and its application in finance
Use the binomial probability distribution to find
probabilities
Describe when to apply the binomial distribution
Use Poisson discrete probability distributions to
find probabilities
5-2
Definitions
Random Variables
A random variable represents a possible
numerical value from an uncertain event.
Discrete random variables produce outcomes
that come from a counting process (e.g. number
of courses you are taking this semester).
Continuous random variables produce outcomes
that come from a measurement (e.g. your annual
salary, or your weight).
5-3
Definitions
Random Variables
Random
Variables
Ch. 5
Discrete
Random Variable
Continuous
Random Variable
Ch. 6
5-4
Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
5-5
Probability Distribution for a
Discrete Random Variable
A probability distribution (or probability mass function )(pdf)
for a discrete random variable is a...