Stat 321- Day 13

Last Time – Binomial vs. Negative Binomial Binomial random variable P(X=x)=C(n,x)p x (1-p) n-x  X = number of successes in n independent trials where p = probability of success for every trial Negative binomial random variable P(X=x)=C(x+r-1, r-1)(1-p) x p r  X = number of failures before r th success Note: 2-3 office hour cancelled today

Example 1 (g) Let X represent the number of unsuccessful operations in a sample of 10 from a population of 371 P(X > 8) Hypergeometric random variable  Two outcomes  Number of successes in a sample of n selections from a finite population N with M successes vs. P(X > 8) from Bin(n=10, p =.15)

When n is small compared to N the binomial and hypergeometric pmf‘s are quite similar

Example 1: Heart Transplantations

Example: Fumbles

Example: Prussian Horsekicks

Summary Hypergeometric Two possible outcomes, sampling without replacement from finite population Binomial Two possible outcomes, independent trials, constant probability of success, finite number of trials Poisson Number of rare events in a fixed interval Geometric Two possible outcomes, independent trials, constant probability of success, number of failures before first success Negative Binomial Two possible outcomes, independent trials, constant probability of success, number of failures before r th success Bernoulli Experiment

Practice (a) Social security claims  Let X represent the number of disability claims examined.  Hypergeometric with N=10, M=4, n=6.  Want to find P(X=4) (b) Toothpaste preference  Let X represent the number of people interview who don’t prefer Brand A before first success  Negative Binomial with r=1 (geometric), p=.60.  Want to find P(X=5)

Practice Suicide rate  Let X represent the number of suicides in a month  Exact: Binomial with n=400,000 and p=  Want P(X>6)  Since n is very large and p is very small, the distribution of X is approximately Poisson with parameter = np = 4. More suicide  Let X represent number of months with 6 or more suicides, want P(X>2)  p=P(success)=P(six or more suicides in a month) = the answer to (c) – assuming stays constant month to month  So X has a binomial distribution with n=12 and p=.2149

Ch. 3 Responsibilities Be able to recognize a situation as hypergeometric, binomial, negative binomial Be able to define the event of interest and calculate the probability for any of these distributions (include poisson)  Be able to use table, minitab for binomial Be able to use the formulas for these distributions to find E(X) and V(X), SD(X) Be able to set up a “generic” pmf, including E(X), SD(X), cdf (and translation)  At least be aware that E(h(X)) usually isn’t h(E(X)) Exception: h(x) is linear

Lab 5 Let Y represent the magnitude of an earthquake and suppose we want to estimate the probability of an earthquake registers 3.0 or higher. Is Y a discrete random variable?

For Friday Probably useful to bring your text, Day 13  Will look at “special” distributions in lab, return to “basics” on Monday Definitely useful to review some basic integration rules