1 Normal Random Variable Stat 200 Sections 1 - 3 Mon Feb 2, 2015 LQ5 available after class Exam 1 Info is posted (Can list topics for review Session under Which as defined is a binomial random variable? For responses from a random sample of 10 PSU students, the: A. X = number of textbooks bought at the beginning of the semester - discrete, non - binomial B. X = number of students who are on twitter (binomial, how many are on twitter out of 10) 0 - 10/N) C. X = student’s blood level of Vitamin D (ng/ml) - continuous D. X = number of times a student plays a computer game until he or she first wins - WAITING n? E. X = time elapsed between when student sits in the classroom and first checks his or her phone - continuous 3 Conditions: Binomial Distribution 1. n fixed trials 2. Two possible outcomes for each trial (binary data) Success: what you want Failure: what you don’t want 3. Outcomes for each trial are independent (partly linked to randomized trials) 4. Probability of success = ( p ) and will remain essentially constant from trial to trial Note: ideally should replace to meet this condition – but practically do not replace – in these instances it is okay (when you take a small sample relative to population size) Check: if n/N < or equal to .05 it does not matter that you have not replaced Note: may not always know value of p 5. Variable defined as: X = number of successes with a fixed n (ideally, the possible values for this random variable are: 0 to n) Note: original data is categorical 4 Practical Situations: Form a committee - no replacement n N n/N ≤ 0.05 p = 0.5 remains constant? indepen - dent trials binomial? 1. Four students are randomly picked from a class of 10 women and 10 men to serve. 4 20 4/20 = .2 <.05 no no no 2. four students are randomly picked from a class of 100 women and 100 men to serve. 4 200 4/200 = .02 < .05 yes yes, random yes 3. four students are randomly picked from a class of 1000 women and 1000 men to serve. 4 2000 4/2000 = 0.002 < .05 yes yes, random yes Success = WOMAN Let X = number of “ women among the four selections Today’s Goals 5 Recall a countable list (integers) probability distribution function assumes a range of values probability density function 7 Example 1: Discrete Probabilities (previous) k P(X = k) 0 0.1 1 0.3 2 0.4 3 0.2 1.0 pdf X = number of courses taught Questions: What is the probability that X is : • exactly 1? P(x=1) = .3 • greater than 1? P(x>1) = .6 • at least 1? P(x>/=1) = .9 (General) Normal: endless Specific: Standard Normal: How many? values for: μ = 0 σ = 1 μ = ANY value σ = ANY value, except 0 random variable: Z approximate range of values: - 3 to 3 = 1.0