Random Variables and Distributions Random Variables and Distributions Random Variables and Distributions Random Variables and Distributions Random Variables and Distributions 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.
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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.
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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
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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
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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
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A discrete random variable X assumes each of its values with a certain probability

7. Example
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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

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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.
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9. Discrete distributions include:
▸ Binomial
▸ Degenerate
▸ Poisson
▸ Geo-metric
▸ Hypergeometric
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10. 2.4 Continuous distribution
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For any continuous random variable, X, there exists a nonnegative function f(x), called
the probability density function (p.d.f) .

11. Remember
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12. Problem
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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)
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15. Find the CDF of the random variable X with the probability function:
Example
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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)
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20. Problem
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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
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Hussein zayed
Pre-Mag student
Gamal Farouk
Supervisor

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THANKS!
Any questions?
You can find me at:
▸ @husseinzayed11
▸ husseinzayed51@gmail.com