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Random Variables and Distributions
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.
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Overview of random variables and their distributions prepared by Hussein Zayed under supervision of Gamal Farouk.
Random variables defined generally as variable values determined by chance; statistical definition relates sample space elements to real numbers.
Illustrates a balanced coin tossed three times as an example of random variables, showing possible outcomes and related values for number of heads.
Distribution describes probabilities of a random variable; presents probabilities of specific outcomes for coin toss results.
Categorization of random variables into discrete and continuous; discrete is countable while continuous spans a range.
This section introduces discrete random variables and their values based on assigned probabilities.
Example of probability function based on two tosses of a non-balanced coin and the number of heads.
Probability function f(x) gives likelihood of discrete random variable outcomes.
List of types of discrete distributions: Binomial, Degenerate, Poisson, Geometric, and Hypergeometric.
Informs about continuous random variables that use probability density functions (pdfs) to describe outcomes.
Brief interlude reminding participants of important concepts within continuous distributions.
Introduces a problem concerning the temperature reaction error as a continuous random variable with a pdf.
Describes CDF for discrete random variables based on probability functions.
Presents an example calculating CDF values for given discrete random variable outcomes.
Defines CDF related to continuous random variables using their probability density functions.
Asks to find CDF and specific probability using CDF for continuous random variable scenario.
Acknowledges team members Hussein Zayed and supervisor Gamal Farouk for contributions.
Concludes the presentation with a thank you note and offers contact information for further questions.
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Random Variables and Distributions
- 1. Random Variables and Distributions Preparedby : Hussein Zayed Supervisor : Gamal Farouk
- 2. 2.1 Random Variables Generaldefinition 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 coinis 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 ofRandom 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 RandomVariables 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 Adiscrete random variable X assumes each of its values with a certain probability
- 7. Example 7 Experiment: tossing anon-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 valuesof 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 Forany continuous random variable, X, there exists a nonnegative function f(x), called the probability density function (p.d.f) .
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- 12. Problem 12 Suppose that theerror 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 distributionfunction (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 CDFof 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 distributionfunction (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 theerror 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-Magstudent Gamal Farouk Supervisor
- 24. 24 THANKS! Any questions? You canfind me at: ▸ @husseinzayed11 ▸ [email protected]