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Chapter 2

Chapter 2. Random Variable & their Distribution. Illustration. Definition. R.V say X is a function defined over a sample space S, that associates a real number, X(e)=x, whith each possible outcome e in S. Look at example above!. Other Example.

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Chapter 2

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  1. Chapter 2 Random Variable & their Distribution

  2. Illustration

  3. Definition R.V say X is a function defined over a sample space S, that associates a real number, X(e)=x, whith each possible outcome e in S Look at example above!

  4. Other Example • An experiments involving a sequence of 5 tosses of a coin, the number of Heads in the sequence is a random variable • Two rolls of a die, v.r : The sum of the two rolls The number of sixes in the two rolls The second roll raised to fifth power

  5. Main Concepts Related to RV • A RV is a real valued function of the outcome of the experiment • A function of R.V defines another R.V • A R.V can be conditioned on an event or on another R.V • There is a notion of independence of a R.V from an event or from another R.V

  6. Definition • let us consider functions which take values in the real numbers. • In the coin tossing example, our function might count the number of heads. Call this function R. We can look at the set • If we have chosen the set of events to contain all subsets of , then this set is an event, and we can ask for the probability of {R=6} • The precise relation is that if the model is (,F,Pr) and R:(-,) then for every interval I, {RI}:={w:R(w)I}F

  7. Definition : Function which satisfied Are called (real valued) R.V

  8. Example 1 R is a R. V since for every interval I the set RI is a subset of , and all subsets of  are in F

  9. Example 2 R is not a R.V since R=2={2} is not in F

  10. Discrete R.V • R.V si discrete if its range is finite or at most countably infinite • Definition : If the set of all possible values of a R.V X is a countable set, then X called a discrete R.V f(x)=P[X=x], called the discrete probability density function (discrete pdf)

  11. Definition

  12. Example 1

  13. Example 2 • The experiment consist of two independent tosses of a fair coin, let X be the number of heads obtained, then the pdf of X is :

  14. Example 3 If Then find c!

  15. EXERCISE

  16. Cummulative Density Function Definition

  17. Theorem A function F(x) is a CDF for some R.V X if and only if it satisfies the following properties :

  18. Example 1

  19. Example 2 Suppose that a days production of 850 manufactured parts contains 50 parts that don’t conform to customer requirements. Two parts are selected at random, without replacement, from the batch. Let the random variable X equal the number of nonconforming parts in the sample. What the cdf of X?

  20. Exercise

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