Random Variables and Distributions
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The document explains the concept of random variables and their distributions, detailing definitions, types, and examples. It distinguishes between discrete and continuous random variables, outlining their probability functions and cumulative distribution functions (CDFs). Additionally, it includes specific examples and problems related to the probability distributions of random variables.
Random Variables and Distributions
- 1. Random Variables and Distributions Prepared by : Hussein Zayed Supervisor : Gamal Farouk
- 2. 2.1 Random Variables General definition A variable whose value is unknown or with a variable value by chance, it is not fixed to a specific value. 2 Statistical definition A random variable X is a function that associates each element in the sample space with a real number (i.e., X : S R.)
- 3. Example A balanced coin is tossed three times, then the sample space consists of eight possible outcomes. Let X the random variable for the number of heads observed. 3 Sample space ( 2^3) = {HHH,HHT,HTH,HTT,THT,THH,TTH,TTT} Let X = {0,1,2,3} X=0 ~ {TTT} X=1 ~ {HTT , THT , TTH} X=2 ~ {HHT , HTH , THH} X=3 ~ {HHH}
- 4. 2.2 Distributions of Random Variables 4 If X is a random variable, then the distribution of X is the collection of probabilities P(X ∈ B) for all subsets B of the real numbers. P(X=0) = ⅛ P(X=1) = ⅜ P(X=2) = ⅜ P(X=3) = ⅛ Sample point (outcomes) Assigned Numerical Value (x) P( X=x ) TTT 0 1/8 THT , TTH , TTH 1 3/8 HHT , HTH , THH 2 3/8 HHH 3 1/8
- 5. Types of Random Variables 5 A random variable X is called a discrete random variable if its set of possible values is countable . A random variable X is called a continuous random variable if it can take values on a continuous scale or range. Discrete x = 5 Continuous 1<=x<5
- 6. 2.3 Discrete Distributions 6 A discrete random variable X assumes each of its values with a certain probability
- 7. Example 7 Experiment: tossing a non-balance coin 2 times independently. Sample space: S={HH, HT, TH, TT} Suppose: P(H)=1/2 P(T) P(H)=1/3 P(T)=⅔ Let X= number of heads
- 8. 8 The possible values of X with their probabilities a The function f(x)=P(X=x) is called the probability function (probability distribution) of the discrete random variable X. 8
- 9. Discrete distributions include: ▸ Binomial ▸ Degenerate ▸ Poisson ▸ Geo-metric ▸ Hypergeometric 9
- 10. 2.4 Continuous distribution 10 For any continuous random variable, X, there exists a nonnegative function f(x), called the probability density function (p.d.f) .
- 11. Remember 11
- 12. Problem 12 Suppose that the error in the reaction temperature, in C, for a controlled laboratory experiment is a continuous random variable X having the following probability density function: 𝇇
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- 14. The cumulative distribution function (CDF), F(x), of a discrete random variable X with the probability function f(x) is given by: 2.5 Cumulative distribution function (CDF) 14
- 15. Find the CDF of the random variable X with the probability function: Example 15 X 0 1 2 F(x) 10/28 15/28 3/28
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- 19. The cumulative distribution function (CDF), F(x), of a continuous random variable X with probability density function f(x) is given by: 2.5 Cumulative distribution function (CDF) 19
- 20. Problem 20 Suppose that the error in the reaction temperature, in C, for a controlled laboratory experiment is a continuous random variable X having the following probability density function: 𝇇 1.Find the CDF 2.Using the CDF, find P(0<X<=1).
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- 23. Team Presentation 23 Hussein zayed Pre-Mag student Gamal Farouk Supervisor
- 24. 24 THANKS! Any questions? You can find me at: ▸ @husseinzayed11 ▸ husseinzayed51@gmail.com