Probability, Random Variables and Expectations Exercises Exercise 1.1. Prove that E [a + b X ] = a + b E [X ] when X is a continuous random variable. Exercise 1.2. Prove that V [a + b X ] = b 2 V [X ] when X is a continuous random variable. Exercise 1.3. Prove that Cov [a + b X , c + d Y ] = b d Cov [X , Y ] when X and Y are a continuous random variables. Exercise 1.4. Prove that V [a + b X + c Y ] = b 2 V [X ] + c 2 V [Y ] + 2b c Cov [X , Y ] when X and Y are a continuous random variables
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probability was obtained by adding the probability, P(x), from the previously itemized probabilities where the cumulative summation of a probability is always equal to one (1) or 100%. A random number formula, =RAND(), was plugged into the Microsoft Excel desired cell, in this situation, (H4), which generated a random range of numbers that are greater than or equal to zero and less than one. The interim time between breakdowns were achieved simply by soliciting the experience several staff members
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the four rolls, and then we could write the expression X 1 + X 2 + X 3 + X4 for the sum of the four rolls. The Xi ’s are called random variables. A random variable is simply an expression whose value is the outcome of a particular experiment. Just as in the case of other types of variables in mathematics, random variables can take on different values. Let X be the random variable which represents the roll of one die. We shall assign probabilities to the possible outcomes of this experiment. We do
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OF SUCCESS IN A VERY SMALL INTERAVAL IS CONSTANT. • THE PROBABILITY OF HAVING MORE THAN ONE SUCCESS IN THE ABOVE REFERRED SMALL TIME INTERVAL IS VERY LOW. • THE PROBABILITY OF SUCCESS IS INDEPENDENT OF t FOR THE TIME INTERVAL(t ,t+dt) . 5. Expected Value or Mathematical Expectation of a random variable may be defined as the sum of the products of the different values taken by the random variable and the corresponding probabilities. Hence if a random variable X takes
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dt 2 2 1 if X~N , 2 and Z= then Z~N 0,1 X- 1 Mathematical Expectation: Given a random variable X and its pdf f x we define the expectation of the function g X to be the integral E g X g x f x dx Note that g X is also a random variable The Moment Generating Function (MGF) The MGF of a random variable X is a function of t denoted by M X t E e xt which is an expectation MGF of normal If X ~ N
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Additional information, including supplemental material and rights and permission policies, is available at http://ite.pubs.informs.org. Vol. 9, No. 1, September 2008, pp. 1–9 issn 1532-0545 08 0901 0001 informs ® doi 10.1287/ited.1080.0014 © 2008 INFORMS INFORMS Transactions on Education Using Simulation to Model Customer Behavior in the Context of Customer Lifetime Value Estimation Shahid Ansari, Alfred J. Nanni Accounting and Law Division, Babson College, Wellesley, Massachusetts
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Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #07 Random Variables So, far we were discussing the laws of probability so, in the laws of the probability we have a random experiment, as a consequence of that we have a sample space, we consider a subset of the, we consider a class of subsets of the sample space which we call our event space or the events and then we define a probability function on that. Now, we consider
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1.10 Sampling Distributions The main objective of most statistical inference is to draw conclusion about the population parameters based on samples studies that is quite small in comparison to the size of the population. In order that conclusion of sampling theory and statistical inference valid, samples must be chosen so as to the representation of a population. For example, Television executives want to know the proportion of television viewers who watch that network’s program. Particularly
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.2 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Variance and Standard Deviation . . . . . . . . . . . . . . 17 3.2.4 “Shortcuts” for Binomial Random Variables . . . . . . . . 18 1 4 Probability Distributions 19 4.1 Binomial Distributions
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CHAPTER 1 : INTRODUCTION 1.1. BACKGROUND OF STUDY Life at the turn of the 21st century are exceptionally testing and not the us effectively we anticipated. Besides, the monetary emergency that is hitting the world these days is no special case for Malaysia likewise influences somewhat by ordinary life in life. As an aftereffect of this, of numerous who wander into the business to oblige the minimal present as a consequence of the present downturn now including understudies. This study is to
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