1. Discrete Random Variables Chapter 4 Objectives 1

2. The student will be able to  Recognize and understand discrete probability distribution functions, in general.  Recognize the Binomial probability distribution and apply it appropriately.  Calculate and interpret expected value (average). 2

3.  Types  General  Binomial  Poisson (not doing)  Geometric (not doing)  Hypergeometric (not doing)  Calculator becomes major tool 3

4.  Probability Distribution Function (PDF)  Characteristics each probability is between 0 and 1, inclusive the sum of the probabilities is 1  An edit of the Relative Frequency Table where the Rel Freq column is relabeled P(X) and we drop the Freq and Cum Freq columns  Calculated from the PDF Mean (expected value) Standard Deviation 4 An example

5.  Characteristics each probability is between 0 and 1, inclusive the sum of the probabilities is 1  fixed number of trials  only 2 possible outcomes  for each trial, probabilities, p and q, remain the same ( p + q = 1)  Other facts  X ~ B(n, p)  X = number of successes  n = number of independent trials  x = 0,1,2,…,n  µ = np  σ = sqrt(npq) 5 Problem 8

6.  What the calculator can do  Find P(X = x) Binompdf(n, p, x)  Find P(X < x) Binomcdf(n, p, x)  What the calculator needs help with  Find P(X < x) = P(X < x-1) Binomcdf(n, p, x-1)  Find P(X > x) = 1 – P(X < x) 1 – Binomcdf(n, p, x)  Find (X > x) = 1 – P(X < x-1) 1 – Binomcdf(n, p, x-1) 6

7. Continuous Random Variables Chapter 5 Objectives 7

8. The student will be able to  Recognize and understand continuous probability density functions in general.  Recognize the uniform probability distribution and apply it appropriately.  Recognize the exponential probability distribution and apply it appropriately. 8

9.  Types  Uniform  Exponential  Normal  Characteristics  Outcomes cannot be counted, rather, they are measured  Probability is equal to an area under the curve for the graph.  Probability of exactly x is zero since there is no area under the curve  PDF is a curve and can be drawn so we use f(x) to describe the curve, I.E. there is an equation for the curve 9

10.  X = a real number between a and b  X ~ U(a, b)  µ = (a+b)/2  σ = sqrt((b-a) 2 /12)  Probability density function  f(x) = 1/(b – a)  To calculate probability find the area of the rectangle under the curve  P (X < x) = (x - a)*f(x)  P (X > x) = (b – x)*f(x)  P (c < X < d) = (d – c)*f(x)  (we are not doing conditional probability) 10

11.  Example - The amount of time a car must wait to get on the freeway at commute time is uniformly distributed in the interval from 1 to 6 minutes. X = the amount of time (in minutes) a car waits to get on the freeway at commute time 1 < x < 6 X ~ U(1, 6) µ = (6 + 1)/2 = 3.50 σ = sqrt((6 – 1) 2 /12) = 1.44 11

12.  What is the probability a car must wait less than 3 minutes? Draw the picture to solve the problem.  P(X < 3) = ____________  P(2.5 < x < 5.6)  Find the 40 th percentile.  The middle 60% is between what values? 12

13.  X ~ Exp(m)  X = a real number, 0 or larger.  m = rate of decay or decline  Mean and standard deviation µ = σ = 1/m therefore m = 1/µ  PDF f (x) = me^(-mx)  Probability calculations P (X < x) = 1 – e^(-mx) P (X > x) = e^(-mx) P (c < X < d) = e^(-mc) – e^(-md)  Percentiles k = (LN(1-AreaToThe Left))/(-m) 13

14.  An example - Count change.  Calculate mean, standard deviation and graph  X = amount of change one person carries  0 < x < ?  X ~ Exp( m )  µ = σ = 1/ m  Find P(X $1.50), P($1.50 < X < $2.50), P(X < k) = 0.90 14

15. The Normal Distribution Chapter 6 Objectives 15

16. The student will be able to  Recognize the normal probability distribution and apply it appropriately.  Recognize the standard normal probability distribution and apply it appropriately.  Compare normal probabilities by converting to the standard normal distribution. 16

17.  The Bell-shaped curve  IQ scores, real estate prices, heights, grades  Notation  X ~ N(µ, σ )  P(X x), P(x 1 < X < x 2 )  Standard Normal Distribution  z-score Converts any normal distribution to a distribution with mean 0 and standard deviation 1 Allows us to compare two or more different normal distributions z = (x - µ)/ σ 17 Comparing

18.  Calculator  Normalcdf(lowerbound,upperbound,µ, σ) if P(X < x) then lowerbound is -1E99 if P(X > x) then upperbound is 1E99  percentiles invNorm(percentile,µ,σ) 18 example

19. The Central Limit Theorem Chapter 7 Objectives 19

20. The student will be able to  Recognize the Central Limit Theorem problems.  Classify continuous word problems by their distributions.  Apply and interpret the Central Limit Theorem for Averages 20

21.  Averages  If we collect samples of size n and n is “large enough”, calculate each sample’s mean and create a histogram of those means, the histogram will tend to have an approximate normal bell shape.  If we use  X = mean of original random variable X, and  X = standard deviation of original variable X then 21

22.  Demonstration of conceptconcept  Calculator  still use normalcdf and invnorm but need to use the correct standard deviation.  Normalcdf(lower, upper,  X,  X /sqrt(n))  Using the conceptconcept 22

23.  What’s fair game  Chapter 4 Chapter 4  Chapter 5 Chapter 5  Chapter 6 Chapter 6  Chapter 7 Chapter 7  21 multiple choice questions  The last 3 quarters’ exams3 quarters’  What to bring with you  Scantron (#2052), pencil, eraser, calculator, 1 sheet of notes (8.5x11 inches, both sides) 23