Stat310
     Mean, Variance and Distributions


                            Hadley Wickham
Saturday, 24 January 2009
1. Recap: random variables & pmf
                2. Expectation
                3. Mean & variance
                4. Meet...
Random variable
                    A random variable is a random
                    experiment with a numeric sample
   ...
Discrete r.v.


                    A discrete random variable has a
                    countable sample space, typically...
pmf
                    Every random variable has an
                    associated probability mass function
            ...
To be a pmf

                    A function must satisfy two properties:
                    • f(x) ≥ 0, for all x
       ...
x   f(x)   x   f(x)   x   f(x)
                            -1 0.3     10 -0.1    1 0.35
                            0   0....
Notation
                    Normally call pmf f
                    If we have multiple rv’s and want to make
           ...
Notation
                    Can give pmf in two ways:
                    • List of numbers (for small n)
               ...
a)                                              b)
       0.8
                                                            ...
Expectation

                    • Allows us to summarise a pmf with a
                      single number
               ...
Mean
                    • Summarises the “middle” of the
                      distribution
                    • If you ...
Variance

                    • Summarises the “spread” of a
                      distribution
                    • Var[...
Meet the distributions


                    Discrete uniform
                    Bernoulli
                    Binomial
S...
Discrete uniform
                    Equally likely events
                    f(x) = 1/m x = 1, ..., m


                ...
Useful facts

                    Sum of integers from i = 1 to m is




                    Sum of squared integers from ...
Bernoulli distribution
                    Single binary event: either happens (with
                    probability p) or...
Transformations
                    If X ~ Bernoulli(p)
                    What is 1 - X?
                    What is X2?...
Binomial distribution
                    n independent Bernoulli trials with the
                    same probability of ...
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06 Mean Var

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06 Mean Var

  1. 1. Stat310 Mean, Variance and Distributions Hadley Wickham Saturday, 24 January 2009
  2. 2. 1. Recap: random variables & pmf 2. Expectation 3. Mean & variance 4. Meet some random variables Saturday, 24 January 2009
  3. 3. Random variable A random variable is a random experiment with a numeric sample space. (Can make many different random variables from a single random experiment) More formally, a random variable is a function that converts elements of non- numeric sample space to numbers. Saturday, 24 January 2009
  4. 4. Discrete r.v. A discrete random variable has a countable sample space, typically a subset of the integers. Saturday, 24 January 2009
  5. 5. pmf Every random variable has an associated probability mass function (pmf). The random variable says what is possible. (the sample space) The pmf says how likely each possibility is. (the probability) Saturday, 24 January 2009
  6. 6. To be a pmf A function must satisfy two properties: • f(x) ≥ 0, for all x • ∑ f(x) = 1 Saturday, 24 January 2009
  7. 7. x f(x) x f(x) x f(x) -1 0.3 10 -0.1 1 0.35 0 0.3 20 0.9 2 0.25 2 0.3 30 0.2 3 0.2 4 0.1 x f(x) x f(x) 5 0.1 5 1 10 0.1 20 0.9 30 0.2 Saturday, 24 January 2009
  8. 8. Notation Normally call pmf f If we have multiple rv’s and want to make clear which pmf belongs to which rv, we write: fX(x) fY(y) fZ(z) for X, Y, Z f1(x) f2(x) f3(3) for X1, X2, X3 Saturday, 24 January 2009
  9. 9. Notation Can give pmf in two ways: • List of numbers (for small n) • Function (for large n) These are equivalent! Also useful to display visually, with a bar plot (not a histogram: the book is wrong!) Saturday, 24 January 2009
  10. 10. a) b) 0.8 1.0 0.6 0.8 0.6 0.4 f(x) f(x) 0.4 0.2 0.2 0.0 0.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 x x c) d) 0.5 0.4 0.4 0.3 0.3 0.2 f(x) f(x) 0.2 0.1 0.1 0.0 0.0 −0.1 1 2 3 4 5 1 2 3 4 5 x x Saturday, 24 January 2009
  11. 11. Expectation • Allows us to summarise a pmf with a single number • Definition • Properties Saturday, 24 January 2009
  12. 12. Mean • Summarises the “middle” of the distribution • If you imagine the number line as a beam with weights of f(x) at position x, the balance point is the mean • mean = E(X) Saturday, 24 January 2009
  13. 13. Variance • Summarises the “spread” of a distribution • Var[X] = E[ (X - E[X])2] = ... • Expected squared distance from centre • sd[X] = Var[X]0.5 Saturday, 24 January 2009
  14. 14. Meet the distributions Discrete uniform Bernoulli Binomial Saturday, 24 January 2009
  15. 15. Discrete uniform Equally likely events f(x) = 1/m x = 1, ..., m X ~ DiscreteUniform(m) What is the mean? What is the variance? Saturday, 24 January 2009
  16. 16. Useful facts Sum of integers from i = 1 to m is Sum of squared integers from i = 1 to m is Saturday, 24 January 2009
  17. 17. Bernoulli distribution Single binary event: either happens (with probability p) or doesn’t happen X ~ Bernoulli(p) What is the mean of X? What is the variance of X? Saturday, 24 January 2009
  18. 18. Transformations If X ~ Bernoulli(p) What is 1 - X? What is X2? (Think about X, not f(x)) Saturday, 24 January 2009
  19. 19. Binomial distribution n independent Bernoulli trials with the same probability of success. X is the number of success X ~ Binomial(n, p) What is the mean? What is the variance? Better tools? Saturday, 24 January 2009

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