Advertisement

PRINTABLE FOR KIDS

XII (12) HSC

XI (11) FYJC
X (10) SSC
9TH 5TH 6TH 7TH 8TH
BOARD SOLUTIONS: 2019 2020 New Time table
ESSAYS DIALOGUE EXPANSION SPEECH LETTERS GRAMMAR WRITING SKILLS INFORMATION-TRANSFER LEAFLET REPORT APPEAL INTERVIEW VIEW AND COUNTERVIEW DATA INPUT SHEET OTHER BOARDS LATEST NEWS  PRIVACY DISCLAIMER
TAMIL-NADU: 8TH 9TH 10TH 11TH 12TH சமையல் மற்றும் சினிமா அ முதல் ஃ வரை 

The Distribution of a Random Variable

In this course, 95% of our efforts will be devoted to events that have numeric outcomes (e.g., the return on an asset, the selling price of a house, the monthly demand for a product or service, etc.).  Probabilistic events that have numerical outcomes are called random variables. 

To describe a random variable completely, we need to know two things: (1) each possible outcome; and (2) the probability of each outcome.  If we have a complete description of both, then we say we know the random variable’s distribution.  There are two general types of random variables (or distributions) encountered in this course: discrete and continuous.

A discrete random variable is one whose outcomes can be listed (like the roll of a die).  A continuous random variable is one whose outcomes are so numerous they cannot be listed.  An example of a continuous random variable would be the time elapsed between customers entering a retail establishment.  If measured to infinitesimal accuracy, one could not list all of the possibilities.  However, if we only measured elapsed times to the nearest second (or minute), then the distribution of elapsed times would be discrete.  In practice, continuous distributions are often used as approximations to discrete distributions in instances where the number of possible outcomes is so large that a continuous distribution makes the analysis easier. 


Example: A Discrete Distribution.  Define a random variable whose value is the sum of the dots obtained from rolling a pair of dice. Construct the probability distribution for this random variable.

           
Outcomes













Probabilities












                       


We frequently summarize information for a discrete random variable by means of its probability histogram.  The probability histogram is simply a visual display of the outcomes (plotted along the x-axis), and their associated probabilities, which are represented by bars (graphed along the y-axis).



Measures of a Distribution: Expectation and Variance (Discrete Case)

The expected value or mean of a discrete random variable (“r.v.” for short) X is denoted by E(X) or  μ and given by the formula μ =E(X) = Σxipi


The expected value is the “theoretical” average obtained by weighting each outcome by its respective probability and then summing.  For the sum of the dice we have

            Possible values (xi)          Probability (pi)               Product (xi pi)

                        2                                  1/36                             2/36
                        3                                  2/36                             6/36
                        4                                  3/36                             12/36
                        5                                  4/36                             20/36
                        6                                  5/36                             30/36
                        7                                  6/36                             42/36
                        8                                  5/36                             40/36
                        9                                  4/36                             36/36
                        10                                3/36                             30/36
                        11                                2/36                             22/36
                        12                                1/36                             12/36
                                                                                                252/36 = 7     (μ =E(X))


Plot the value 7 on the probability histogram.  Observe that E(X) is a measure of centrality.

Example.  You sell big electric motors.  During a given week, demand for your 100-hp motor is 0, 1, or 4 (4 come on a palette).  The distribution is described below.

Demand
0
1
4
Probability
.45
.40
.15

What is your expected demand for a week?
E(X) =  (0)(.45)+(1)(.4)+(4)(.15) = 1.00

Another measure of interest is the expected value of the expression (X – E(X)2), called the variance of X, and given by the formula
Var(X) = E(X) – [E(X)]2

The variance also goes by the Greek letter σ2.  The formula looks bad, but a few simple examples will clarify its calculation and help us understand what it tells us.  Note that the mean of X is needed before computing the variance.  Recall that pi is the probability that X takes on the value xi.

PDF FILE TO YOUR EMAIL IMMEDIATELY PURCHASE NOTES & PAPER SOLUTION. @ Rs. 50/- each (GST extra)

SUBJECTS

HINDI ENTIRE PAPER SOLUTION

MARATHI PAPER SOLUTION
SSC MATHS I PAPER SOLUTION
SSC MATHS II PAPER SOLUTION
SSC SCIENCE I PAPER SOLUTION
SSC SCIENCE II PAPER SOLUTION
SSC ENGLISH PAPER SOLUTION
SSC & HSC ENGLISH WRITING SKILL
HSC ACCOUNTS NOTES
HSC OCM NOTES
HSC ECONOMICS NOTES
HSC SECRETARIAL PRACTICE NOTES

2019 Board Paper Solution

HSC ENGLISH SET A 2019 21st February, 2019

HSC ENGLISH SET B 2019 21st February, 2019

HSC ENGLISH SET C 2019 21st February, 2019

HSC ENGLISH SET D 2019 21st February, 2019

SECRETARIAL PRACTICE (S.P) 2019 25th February, 2019

HSC XII PHYSICS 2019 25th February, 2019

CHEMISTRY XII HSC SOLUTION 27th, February, 2019

OCM PAPER SOLUTION 2019 27th, February, 2019

HSC MATHS PAPER SOLUTION COMMERCE, 2nd March, 2019

HSC MATHS PAPER SOLUTION SCIENCE 2nd, March, 2019

SSC ENGLISH STD 10 5TH MARCH, 2019.

HSC XII ACCOUNTS 2019 6th March, 2019

HSC XII BIOLOGY 2019 6TH March, 2019

HSC XII ECONOMICS 9Th March 2019

SSC Maths I March 2019 Solution 10th Standard11th, March, 2019

SSC MATHS II MARCH 2019 SOLUTION 10TH STD.13th March, 2019

SSC SCIENCE I MARCH 2019 SOLUTION 10TH STD. 15th March, 2019.

SSC SCIENCE II MARCH 2019 SOLUTION 10TH STD. 18th March, 2019.

SSC SOCIAL SCIENCE I MARCH 2019 SOLUTION20th March, 2019

SSC SOCIAL SCIENCE II MARCH 2019 SOLUTION, 22nd March, 2019

XII CBSE - BOARD - MARCH - 2019 ENGLISH - QP + SOLUTIONS, 2nd March, 2019

HSC Maharashtra Board Papers 2020

(Std 12th English Medium)

HSC ECONOMICS MARCH 2020

HSC OCM MARCH 2020

HSC ACCOUNTS MARCH 2020

HSC S.P. MARCH 2020

HSC ENGLISH MARCH 2020

HSC HINDI MARCH 2020

HSC MARATHI MARCH 2020

HSC MATHS MARCH 2020


SSC Maharashtra Board Papers 2020

(Std 10th English Medium)

English MARCH 2020

HindI MARCH 2020

Hindi (Composite) MARCH 2020

Marathi MARCH 2020

Mathematics (Paper 1) MARCH 2020

Mathematics (Paper 2) MARCH 2020

Sanskrit MARCH 2020

Sanskrit (Composite) MARCH 2020

Science (Paper 1) MARCH 2020

Science (Paper 2)

Geography Model Set 1 2020-2021


MUST REMEMBER THINGS on the day of Exam

Are you prepared? for English Grammar in Board Exam.

Paper Presentation In Board Exam

How to Score Good Marks in SSC Board Exams

Tips To Score More Than 90% Marks In 12th Board Exam

How to write English exams?

How to prepare for board exam when less time is left

How to memorise what you learn for board exam

No. 1 Simple Hack, you can try out, in preparing for Board Exam

How to Study for CBSE Class 10 Board Exams Subject Wise Tips?

JEE Main 2020 Registration Process – Exam Pattern & Important Dates


NEET UG 2020 Registration Process Exam Pattern & Important Dates

How can One Prepare for two Competitive Exams at the same time?

8 Proven Tips to Handle Anxiety before Exams!

BUY FROM PLAY STORE

DOWNLOAD OUR APP

HOW TO PURCHASE OUR NOTES?

S.P. Important Questions For Board Exam 2022

O.C.M. Important Questions for Board Exam. 2022

Economics Important Questions for Board Exam 2022

Chemistry Important Question Bank for board exam 2022

Physics – Section I- Important Question Bank for Maharashtra Board HSC Examination

Physics – Section II – Science- Important Question Bank for Maharashtra Board HSC 2022 Examination

Important-formula


THANKS