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Advanced Concepts in Applied Probability: Probability Mass Functions and Expectations

This lecture delves into key concepts in applied probability, focusing on Probability Mass Functions (PMFs) and Conditional PMFs, with examples illustrating their applications in statistics. The discussion includes the relationship between electric load and population dynamics in Tucson, AZ, highlighting the role of missing variables like temperature. Further topics cover cumulative density functions, the expected value, and variance calculations, including practical examples such as Bernoulli trials and exponential probability density functions. Dive deep into the intricacies of probability and statistics!

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Advanced Concepts in Applied Probability: Probability Mass Functions and Expectations

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  1. Applied Probability Lecture 3 Rajeev Surati

  2. Agenda • Statistics • PMFs • Conditional PMFs • Examples • More on Expectations • PDFs • Introduction • Cumalative Density Functions • Expectations, variances

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

  4. 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

  5. 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.

  6. 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

  7. Conditional PMF • and independence Implies for all x and y Example: derive PMFs

  8. Expectations continued Expectation of g(x,y) Compute E(x+y) Compute

  9. 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

  10. 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.

  11. Same old set of rules except…

  12. Some Example Events • X<= 2 • 1 <= x <= 10

  13. An Example • Exponential pdf

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