further statistics

• to find E(X) from a table, multiply the x values by the probability
• what is the formula for the variance in terms of E(X)?

  Var(X) = E(X^2) - E(X)^2
• what is a probability distribution?

  a table of values and probabilities
• when adding poisson distributions, the mean is the sum of the means of the two distributions
• to approximate binomial to poisson: X ~ Po(np)
• for a geometric distribution, the formula for the cumulative distribution for P(X≥x) is (1 - p)^(x - 1)
• for a geometric distribution, the formula for the cumulative distribution for P(X≥x) is 1 - (1 - p)^x
• the formula for applying the central limit theorem is X ~ N(mean, variance/n)
• the formula for a chi squared test is the sum of (observed frequency)^2/(expected frequency) - number of trials
• the degrees of freedom = number of cells - number of parameters
• the power of a test = 1 - P(type 2 error)
• for a contingency table, expected frequency = (row total x column total)/grand total
• Var(aX + b) = a^2 Var(X)
• E(aX + b) = aE(X) + b
• what are the conditions required to use a poisson distribution?

  1. events occur singly in time
  2. events occur at a constant rate
  3. events occur independently
• in a poisson distribution, the variance and mean are the same
• the conditions for approximating binomial with poisson are:

  • n is large
  • p is small
• for a geometric distribution to be a suitable model, trials must be independent, the probability of success is the same for each trial.
• the mean of a geometric distribution is 1/p
• the variance of a geometric distribution is (1-p)/p^2
• negative binomial is written as X ~ NB(r, p), where r = the number of successes required
• mean of a negative binomial distribution = r/p
• the variance of a negative binomial distribution is r(1 - p)/p^2
• a poisson hypothesis test tests for the mean
• a hypothesis test for a geometric distribution tests for the number of trials required for one success
• when using poisson approximation for a binomial during a hypothesis test, you write the hypotheses using the binomial distribution
• typically, if a question about a hypothesis test or distribution asks you to 'estimate' or something to do with the 'mean' or 'average', it might be a central limit theorem question
• in a chi squared hypothesis test, the null hypothesis is that the distribution is a good fit, and the alternative hypothesis is that the distribution is not a good fit
• process for a goodness of fit test:

  1. state your hypotheses
  2. calculate the degrees of freedom
  3. calculate the expected frequencies for each outcome
  4. create a table for the expected and observed frequencies
  5. if any expected frequencies are less than 5, group them until they are greater than 5
  6. calculate the chi squared value, you must list at least 3 terms of the formula
  7. check against the critical value from the tables
• probability functions:

  Gx(t) = Σ P(X = x) t^x
• to prove one of the given PGFs (from formula book), you should use the regular probability generating function formula and use the formula for P(X=x) for that particular distribution
• expected value of X in terms of a probability generating function: E(X) = G ' x (1)

  (in formula book)
• substituting in t = 0 into the nth derivative of a PGF will give you P(X = N)
• a type 1 error is when the null hypothesis is rejected when it is actually true. it is the same as the the actual significance level.
• a type 2 error is when you incorrectly accept the null hypothesis
• to find the probability of a type 2 error, use the true value (given in the question) and use your calculator to find the probability of the original acceptance region with the new parameters.
• the size of a test is equal to the probability of a type 1 error
• power of a test = 1 - P(type 2 error)
• the power of a test is the probability of correctly rejecting the null hypothesis
• when you don't have the true value to calculate the power of a test, you can create a power function using the formula for P(X=x) (in formula book) in terms of a variable, such as the mean or probability