Probability and Statistics, Midterm Nov. 3, 2011 Instructions. The exam is from 3:40-5:30. Simplify your answer as much as you can, for instance, x 1 x+1 should be simpli…ed to x 1: But the answer does not need to be numbers if the caculation is complicated, e5 for example, you can leave 20! in your answer. 1. (30 points) (a) One of three students, A, B and C, will get the prize for being the "best student in Probability and Statistics". The night before the prize is announced, student A …nds the professor and asks, "No matter I get the prize or not, we know that at least one of B and C does not get it. Would you please tell me one of them who does not win it? I’ like to prepare a gift for her." The professor refuses, d "No, I cannot give you any hint about the probability that you win the prize." But student A insists, "Think about it again, professor. Telling me one of B and C who does not win the prize won’ give me t any clue about my chances of winning it." Do you agree with A’ claim? Why? s Answer: His claim is correct under certain assumptions. Suppose that the prior probability of each one winning the prize is 1=3; then the question is exactly the same as the "two sheep, one car" game that we introduced in the class. In fact, if the professor tells A that B does not win, the posterior can be caculated as following: Prior A wins B wins C wins 1=3 1=3 1=3 Conditional probability that the professor says "B" 1=2 0 1 Posterior (1=3) (1=2) = 1=3 (1=3) (1=2) + (1=3) 1 0 (1=3) 1 = 2=3 (1=3) (1=2) + (1=3) 1 2

Thus the conditional probability that A wins is equal to the unconditional probability. (b) There are two villages. In the …rst village, each couple is allowed to have one child at most. In the other one there is no such a constraint. Suppose all parents want boys, so that in the second village each couple will keep trying until they get a boy. In the end, will the Boy/Girl ratios in the two villages be extremely di¤erent? Reason your answer. Answer: Intuitively, if each child independently has probability to be a boy and 1 probability to be a girl, then no matter how many children a couple already have, the probability of the next child being a boy is still : Comparing the two villages, although the numbers of children (per family) may be very di¤erent, the probabilities of each child being a boy are both . Thus the percentage of boys is approximately in each village. The boy/girl ratio should be approximately = (1 ): Formally, let random variable X be the number of girls in a family in the second village. Then under conventional assumptions, X has a geometric distribution (a special negative binomial distribution with r = 0). The p.d.f. of X is x f (x) = (1 ) and the expectation of X is E (X) = 1 X

x (1 (1

)

x

x=0

) E (X) = =
1 X

) E (X) = )
x 1 X

x=0

1 X

x (1 )

)
x

x

x=0

1 X

x (1

)

x+1

x (1 (1 )

(x

1) (1 )
1

) E (X) = (1

x=0 1 X

x=1 x

=

x=1

1 (1 1 (1

)

(1

)

=1

)= 1

Suppose that there are n families in the second village and let Xi be the number of girls in family i. Then Xi are independent and have the same distribution as X: Since the number of boys per family in the second village is 1; the boy/girl ratio of second village is Pn 1 1 Pni=1 = Pn Xi i=1 i=1 Xi n Using the law of large numbers (which we’ learn soon), the ratio converges to ll 1

almost surely.

(c) 11 cards, one letter on each, compose the word "probability". Now randomly pick 7 cards one by one and without replacement. What is the probability that the 7 letters on the 7 cards sequentially form the word "ability"? What is the probability that the 7 cards can be used (by reordering) to form the word "ability". Answer: (i) In total there are "ability", there are 11! (11 7)!

possible ordered sequences of the 7 cards, and for the them to be 1 2 2 1 (2 1) 1 1 = 4

...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,...

...manipulation of raw
data into meaningful information are the heart of data analysis. Data analysis includes data
description, data inference, the search for relationships in data and dealing with uncertainty
which in turn includes measuring uncertainty and modelling uncertainty explicitly.
In addition to data analysis, other decision making techniques are discussed. These techniques
include decision analysis, project scheduling and network models.
Chapter 1 illustrates a number of ways to summarise the information in data sets, also known as
descriptive statistics. It includes graphical and tabular summaries, as well as summary measures
such as means, medians and standard deviations.
Uncertainty is a key aspect of most business problems. To deal with uncertainty, we need a basic
understanding of probability. Chapter 2 covers basic rules of probability and in Chapter 3 we
discuss the important concept of probability distributions in some generality.
In Chapter 4 we discuss statistical inference (estimation), where the basic problem is to estimate
one or more characteristics of a population. Since it is too expensive to obtain the population
information, we instead select a sample from the population and then use the information in the
sample to infer the characteristics of the population.
In Chapter 5 we look at the topic of regression analysis which is used to study relationships
between variables....

...APPLIED PROBABILITY AND STATISTICS
APPLIED PROBABILITY AND STATISTICS
DEPARTMENT OF COMPUTER SCIENCE
DEPARTMENT OF COMPUTER SCIENCE
STATISTICAL DISTRIBUTION
STATISTICAL DISTRIBUTION
SUBMITTED BY –
PREETISH MISHRA (11BCE0386)
NUPUR KHANNA (11BCE0254)
SUBMITTED BY –
PREETISH MISHRA (11BCE0386)
NUPUR KHANNA (11BCE0254)
SUBMITTED TO –
PROFESSOR
SUJATHA V.
SUBMITTED TO –
PROFESSOR
SUJATHA V.
ACKNOWLEDGEMENT –
ACKNOWLEDGEMENT –
First and foremost we like to thank our supervisor of the project Mrs Sujatha V. for her valuable guidance an advice. She inspired us greatly to work in this project. Her willingness to motivate us contributed majorly in this project. We also would like to thank her for showing us some example that related to our project.
Besides, we would also like to thank VIT University for providing us with a good environment and facilities to complete this project. Also, we would like to thank school of computer science (SCSE) of VIT University, for offering this subject and computing project. It has given us the opportunity to participate and learn about various methods of calculating statistical distribution.
Finally, an honourable mention goes to my team for completing this project. Without helps of particular mentioned above, we would have faced many difficulties while doing this...

...Chapter 1
The Problem and Its Background
Introduction
Changes are permanent thing on earth. Are the people is ready enough to accept the changes on the educational system? The current opening of classes here in the Philippines usually starts from June to March but our lawmakers want to amend the opening of classes. The existing school calendar which spans from June to March is often disrupted as destructive typhoons plague the region during the rainy season that’s why our lawmakers decided to move the opening of classes from September through May to avoid numerous class suspensions and serve to protect the students during strong typhoons. The Department of Education said that it is open to the proposal by some sectors, including lawmakers to move the opening of classes but they want to ensure the comfort of the students in school and stresses the need for a comprehensive study.
While the Department of Education is open on the proposal, some did not welcome this idea. An initial survey on the matter conducted way back in 2009. On the respondents, 66 percent were against the move while 34 percent were in favor. [1] Also, another ground for rejecting the proposal is the traditions celebrated during summer and being not conducive in learning because of hot weather during March. The delay in suspension of classes is one of the reasons why our lawmakers and other sectors in the community urge to move the classes from June to September. Unexpected suspension of classes is...

...MM207 Statistics
Unit IV Mid Term Project
1. In the following situation identify the implied population.
A recent report on the weekly news presented the findings of a study on the effectiveness of Onglyza, along with diet and exercise, for treating diabetes.
According to Bennett (2009), a population is defined as “the complete set of people or things being studied” in a statistical study. Given that the information is in relation to finding the success of a drug used to care for diabetes, the population would be individuals who experience diabetes. Therefore, the implied population is the entire individual’s who diabetics are. Individuals who are diabetics are those who were used to test the effectiveness of Onglyza, diet and exercise.
2. In the following scenario identify the type of statistical study that was conducted.
A Gallop poll surveyed 1,018 adults by telephone, and 22% of them reported that they smoked cigarettes within the past week.
I would say that this would be an observational study which is a study when specific characteristics of the subject are observed, but the characteristics are in no way customized by the researcher. The reason I say that this would be an observational study is because the sample population that was studied was not influenced by the researcher themselves. In addition based on the fact that this study was a poll, in which people were asked to answer the questions but no responses were...

...TLFeBOOK
FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS
T.T. Soong State University of New York at Buffalo, Buffalo, New York, USA
TLFeBOOK
TLFeBOOK
FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS
TLFeBOOK
TLFeBOOK
FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS
T.T. Soong State University of New York at Buffalo, Buffalo, New York, USA
TLFeBOOK
Copyright 2004
John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England Telephone ( 44) 1243 779777
Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All R ights R eserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, D esigns and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court R oad, London W1T 4LP, UK, without the permission in writing of the Publisher. R equests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to ( 44) 1243 770620. This publication is...

...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...