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1. Chapter 4

   Discrete Random Variables Chapter 4 Objectives
2. Chapter 4 Objectives 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).
3. Discrete Random Variables Types General Binomial Poisson (not doing) Geometric (not doing) Hypergeometric (not doing) Calculator becomes major tool
4. General Discrete Variables 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 RelFreq column is relabeled P(X) and we drop the Freq and Cum Freq columns Calculated from the PDF Mean (expected value) Standard Deviation An example
5. Binomial 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) Problem 8
6. Using Calculator for Binomial 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)

7. Chapter 5

   Continuous Random Variables Chapter 5 Objectives
8. Chapter 5 Objectives The student will be able to Recognize and understand continuous probability distribution functions in general. Recognize the uniform probability distribution and apply it appropriately. Recognize the exponential probability distribution and apply it appropriately.
9. Continuous Random Variables 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
10. Exponential Distribution 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) = P (X < d) - P (X < c) = 1 - e^(-md) - (1 – e^(-mc) = e^(-mc) – e^(-md) Percentiles k = (LN(1-AreaToThe Left))/(- m)
11. Exponential Distribution 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 < $2.50), P(X > $1.50), P($1.50 < X < $2.50), P(X < k) = 0.90
12. Uniform Distribution 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)
13. Uniform Distribution 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 = 3.5 σ = sqrt((6 – 1)2/12) = 1.4434
14. Uniform Distribution What is the probability a car must wait less than 3 minutes? Draw the picture to solve the problem. P(X < 3) = ____________
15. Uniform DistributionConditional Probability What is probability that a car waits more than 4 minutes given it has already waited more than 3 minutes? P(X >4|X >3) = _____________

16. Chapter 6

    The Normal Distribution Chapter 6 Objectives
17. Chapter 6 Objectives 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.
18. The Normal Distribution The Bell-shaped curve IQ scores, real estate prices, heights, grades Notation X ~ N(µ, σ ) P(X < x), P(X > x), P(x1 < X < x2) 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
19. The Normal Distribution 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)

20. Chapter 7

    The Central Limit Theorem Chapter 7 Objectives
21. Chapter 7 Objectives 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
22. The Central Limit Theorem 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
23. Central Limit Theorem Demonstration of concept Calculator still use normalcdf and invnorm but need to use the correct standard deviation. Normalcdf(lower, upper,X,X/sqrt(n)) Using the concept
24. Review for Exam 2 What’s fair game Chapter 4 Chapter 5 Chapter 6 Chapter 7 21 multiple choice questions The last 3 quarters’ exams What to bring with you Scantron (#2052), pencil, eraser, calculator, 1 sheet of notes (8.5x11 inches, both sides)