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2. Random variables

• Introduction
• Distribution of a random variable
• Distribution function properties
• Discrete random variables
• Point mass
• Discrete uniform
• Bernoulli
• Binomial
• Geometric
• Poisson   

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2. Random variables

• Continuous random variables
• Uniform
• Exponential
• Normal
• Transformations of random variables
• Bivariate random variables
• Independent random variables
• Conditional distributions
• Expectation of a random variable
• kth moment

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2. Random variables

• Variance
• Covariance
• Correlation
• Expectation of transformed variables
• Sample mean and sample variance
• Conditional expectation

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Introduction

• Random variables assign a real number to each
• outcome

• Random variables can be
• Discrete if it takes at most countably many
• values (integers).
• Continuous if it can take any real number.

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Distribution of a random variable
Distribution function
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Distribution function properties
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Distribution of a random variable

• For a random variable, we define
• Probability function
• Density function,
• depending on wether is either discrete or
  continuous

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Distribution of a random variable
Probability function
verifies
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Distribution of a random variable
Probability density function
verifies
We have
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Distribution of a random variable
completely determines the distribution of
a random variable.
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Discrete random variables
Point mass
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Discrete random variables
Discrete uniform
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Discrete random variables
Bernoulli
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Discrete random variables
Binomial Successes in n independent Bernoulli
trials with success probability p
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Discrete random variables
Geometric Time of first success in a sequence of
independent Bernoulli trials with success
probability p
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Discrete random variables
Poisson X expresses the number of rare events
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Continuous random variables
Uniform
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Continuous random variables
Exponential
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Continuous random variables
Normal
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Continuous random variables

• Properties of normal distribution
• standard normal
• (ii)
• (iii) independent
  i1,2,...,n

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Transformations of random variables
X random variable with Y r(x)
distribution of Y ? r() is one-to-one r -1().
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Bivariate random variables

• (X,Y) random variables
• If (X,Y) is a discrete random variable
• If (X,Y) is continuous random variable

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Bivariate random variables
The marginal probability functions for X and Y
are
For continuous random variables, the
marginal densities for X and Y are
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Independent random variables
Two random variables X and Y are independent
if and only if for all values x and y.
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Conditional distributions
Discrete variables
Continuous variables
If X and Y are independent
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Expectation of a random variable

• Properties
• (i)
• If are independent then

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Moment of order k
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Variance
Given X with standard
deviation
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Variance

• Properties
• (i)
• If are independent then
• (iii)
• (iv)

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Covariance
X and Y random variables
Properties (i) If X, Y are independent
then (ii) (iii) V(X Y) V(X)
V(Y) 2cov(X,Y) V(X - Y) V(X)
V(Y) - 2cov(X,Y)
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Correlation
X and Y random variables
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Correlation

• Properties
• (i)
• If X and Y are independent then

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Expectation of transformed variables

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Sample mean and sample variance
Sample mean
Sample variance
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Sample mean and sample variance
Properties X random variable
i. i. d. sample, Then (i)
(ii) (iii)
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Conditional expectation
X and Y are random variables Then
Properties
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