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Elements of Distribution Theory Table of contents Random The sample space. Events. Definition of probability Axioms of probability theory Conditional probability Independence Bayesian theory Random Variable Discrete Random Variable Revision and Test 1 Continuous random Variable Probability Mass Function Probability Density Function Cumulative Distribution Function Expectation, Variance and Moment for Discrete random variable> Moment generating function Binomial Distribution Poisson Distribution Test2 Geometric Distribution Hyper- geometric Distribution Negative Binomial Distribution Uniform Distribution Exponential Distribution experiment Normal Distribution Standard Normal Distribution Lesson 1 Random experiment : There are lots of phenomena in nature, like tossing a coin or tossing a die, whose outcomes cannot be predicted with certainty in advance, but the set of all the possible outcomes is known. These are what we call random phenomena or random experiments. Probability theory is concerned with such random phenomena or random experiments. Sample Spaces and Events Sample Spaces Definition: A sample space is a collection of all possible outcomes of a random experiment. We can say "the sample space of a random experiment is a set S that includes all possible outcomes of the experiment"; the sample space plays the role of the universal set when modeling the experiment. For simple experiments, the sample space may be precisely the set of possible outcomes. More often, for complex experiments, the sample space is a mathematically convenient set that includes the possible outcomes and perhaps other elements as well. For example, if the experiment is to throw a standard dice and record the outcome, the sample space is S={1,2,3,4,5,6} , the set of possible outcomes. On the other hand, if the experiment is to capture a small insect and measure its body weight (inmilligrams), we might conveniently take the sample space to be S=[0,∞) , even though most elements of this set are practically impossible (we hope!). Examples: Write the sample space for the following random experiments: 1234567- Tossing a coin. Tossing a die. Tossing a coin twice. Tossing a die twice. Choosing a point from the interval (0; 1). Measuring the lifetime of a light bulb. Keeping on tossing a coin until one gets a Heads. Solutions 1- The sample space is S {H , T }. 2- The sample space is S {1,2,3,4,5,6}. 3- The sample space is S {HH , HT , TH , TT }. 4- The sample space is S {( i, j ) : i, j 1,2,...,6}, which contains 36 elements. 5- The sample space is S = (0; 1). 6- The sample space is S = [0;∞). 7- The sample space of this experiment is {H , TH , TTH , TTTH ,....} . Events Definition: Any subset E of the sample space S is called an event. Certain subsets of the sample space of an experiment are referred to as events. Thus, an event is a set of outcomes of the experiment. Each time the experiment is run, a given event A either occurs, if the outcome of the experiment is an element of A , or does not occur, if the outcome of the experiment is not an element of A . Intuitively, you should think of an event as a meaningful statement about the experiment. In particular, the sample space S itself is an event; by definition it always occurs. At the other extreme, the empty set ∅ is also an event; by definition it never occurs. The Algebra of Events The standard algebra of sets leads to a grammar for discussing random experiments and allows us to construct new events from given events. In the following exercises, suppose that A and B are events. A⊆B if and only if the occurrence of A implies the occurrence of B . A∪B is the event that occurs if and only if A occurs or B occurs. A∩B is the event that occurs if and only if A occurs and B occurs. A and B are disjoint if and only if they are mutually exclusive; they cannot both occur on the same run of the experiment. A∖B is the event that occurs if and only if A occurs and B does not occur. Ac is the event that occurs if and only if A does not occur. c c (A∩ B )∪(B∩ A ) is the event that occurs if and only if one but not both of the given events occurs. Recall that this event is the symmetric difference of A and B , and is sometimes denoted AΔB . c c (A∩B)∪( A ∩ B ) is the event that occurs if and only if both or neither of the given events occurs. Examples: 1- A coin is tossed twice and the outcome of each is recorded, find event that the second toss was a Head? solution the sample space is the set of all possible outcomes. S= {(HH), (HT), (TH), (TT)}. The event that the second toss was a Head is the subset E = {(H, H), (T, H)}. 2- Consider 3 light-bulbs. Our experiment consists of finding out which lightbulb burns out first, and how long (in hours) it takes for this to happen. Solution The sample space is: S = {(i, t) : i ∈ {1, 2, 3}, t ≥ 0} i tells you which one burns out, and t gives how long it lasted, in hours. The event that the 2nd bulb burns out first, and it lasts less than 3 hours is the set E = {(2, t) : t < 3}. Exercise: 1- Find the event of occurrence of " Head" when tossing a coin. 2- The occurrence of an odd number when tossing a dice. 3- The appearance of "Tail" in the second toss when tossing a coin three times.