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Understanding Random Variables and Their Distributions: Definitions and Examples

Explore the concept of random variables (RV) and their distributions, including discrete RV, probability density functions, and cumulative density functions. Learn how RVs are defined and related to experimental outcomes with concrete examples.

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Understanding Random Variables and Their Distributions: Definitions and Examples

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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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