Applied probability lecture 3
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Applied Probability Lecture 3 . Rajeev Surati. Agenda. Statistics PMFs Conditional PMFs Examples More on Expectations PDFs Introduction Cumalative Density Functions Expectations, variances. Statistics. If the number of citizens in a city goes up should the electric load go up?.

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Agenda
Agenda

  • Statistics

  • PMFs

    • Conditional PMFs

    • Examples

    • More on Expectations

  • PDFs

    • Introduction

    • Cumalative Density Functions

    • Expectations, variances


Statistics
Statistics

If the number of citizens in a city goes up should the electric load go up?


Statistics1
Statistics

  • Statistically I can show that in Tucson Arizona the electric load goes up when the number of people goes down when people leave at the end of the winter

  • Does that mean that people leaving caused the rise?

  • The missing variable is temperature


Probability mass functions
Probability Mass Functions

  • Consider which equals probability that the values of x,y are and is often called the compound p.m.f.

    and vis a vis.


An example
An example

  • Show the pmf for p(r,h) of three coin flips, where length of longest run r and # of heads h

  • Show that you can derive a distribution

  • Expected value and variance of r


Conditional pmf
Conditional PMF

  • and independence

    Implies for all x and y

    Example: derive PMFs


Expectations continued
Expectations continued

Expectation of g(x,y)

Compute E(x+y)

Compute


One last pmf example
One last PMF Example

  • Bernoulli Trial 1 if heads, 0 if tails

  • Compute expected value and variance

  • Compute expected value and variance of the sum of n such bernoulli trials


Probability density function
Probability Density Function

  • Here we are dealing with describing a set of points over a continuous range. Since the number of points is infinite we discuss densitiies rather than “masses” or rather PMFs are just PDFs with impulse functions at each discrete point in the PMF domain.



Some example events
Some Example Events

  • X<= 2

  • 1 <= x <= 10


An example1
An Example

  • Exponential pdf


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