Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Lab2 Probability Sampling and Basic Plotting M George Akritas M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Outline Review of Probability Sampling A Special PMF Combinations in R The Binomial Theorem and PMF Sampling from the Binomial PMF Basic Plotting in R The Lab Assignment 2 M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Definition The type of random sampling where different outcomes have different probability of occurring is referred to as probability sampling Probability sampling is always sampling with replacement Example Consider rolling a die twice and record the number of times 6 occurs Describe an equivalent probability sampling experiment Solution The experiment that records the number of 6s can be thought of as a probability sampling experiment from the sample space population S 0 1 2 with probabilities P 0 25 36 P 1 10 36 and P 2 1 36 M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Definition A list of the numerical outcomes in the sample space population and their associated probabilities is be called a probability mass function or pmf for short The pmf of the previous example is x prob 0 25 36 1 10 36 2 1 36 Probability sampling is also called sampling from a pmf M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 I Factorials in R I I I I Combinations in R The Binomial Theorem and PMF Sampling from the Binomial PMF x can be given by factorial x or gamma x 1 Try factorial 3 and gamma 4 Also factorial 4 and gamma 5 Try also factorial 3 5 and gamma 3 5 Combinations in R I I I For the number of combinations kn try factorial 5 factorial 3 factorial 5 3 for 53 or factorial 10 factorial 3 factorial 10 3 for 10 3 For the actual combinations though this is not needed as often try combn 1 5 3 or combn 1 10 3 Note 1 5 gives the same result as seq 1 5 or seq 1 5 or seq 1 5 1 Can also do length combn 1 5 3 3 for 53 M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Combinations in R The Binomial Theorem and PMF Sampling from the Binomial PMF Theorem The Binomial Theorem p q n n n 0 n n 1 1 n 0 n p q p q p q 0 1 n n n X n n k k X n k n k p q p q k k k 0 I I k 0 If p q 1 and p q 0 we get a p m f on S 0 1 n with P k kn p k q n k This is called the Binomial p m f If p q 0 5 P k kn 0 5n is the probability of having k heads in n flips of a fair coin M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 I The Binomial p m f I I I Combinations in R The Binomial Theorem and PMF Sampling from the Binomial PMF Try dbinom 0 10 10 0 5 for all probabilities P k k 0 10 Try also dbinom 4 10 0 5 for P 4 Verify that P 4 obtained above can also be obtained from factorial 10 factorial 4 factorial 10 4 0 5 10 Probability sampling using the Binomial p m f I I I Try sample 0 10 1000 TRUE dbinom 0 10 10 0 5 or p dbinom 0 10 10 0 5 sample 0 10 1000 TRUE p Try it again using object x to store the sample x sample 0 10 1000 TRUE dbinom 0 10 10 0 5 For a summary of the sample do table x or table x 1000 Compare with dbinom 0 10 10 0 5 M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 I Plotting a p m f I I I I I Try plot 0 10 dbinom 0 10 10 0 5 which is the same as plot 0 10 dbinom 0 10 10 0 5 pch 1 or plot 0 10 dbinom 0 10 10 0 5 pch 4 Add color Try plot 0 10 dbinom 0 10 10 0 5 pch 4 col 2 Customize labels plot 0 10 dbinom 0 10 10 0 5 pch 1 col 2 xlab Sample Space ylab Binomial Probabilities Connecting the dots Try lines 0 10 dbinom 0 10 10 0 5 and also lines 0 10 dbinom 0 10 10 0 5 col 3 Bar graph of a p m f I I I Try p dbinom 0 10 10 0 5 barplot p Customize the axes barplot p xlim c 0 12 ylim c 0 0 25 Add color barplot p xlim c 0 12 ylim c 0 0 25 col green M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 I Histogram of Empirical Probabilities I I I Try hist x seq 0 5 10 5 1 Add color hist x seq 0 5 10 5 1 col 5 Check out the site http www harding edu fmccown r for additional still basic plotting examples M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 Do computer activities 1 2 3 in Section 2 8 Email your results together with the plots you can save them as pdf s for example to the TA by Friday 16th of September M George Akritas Lab2 Probability Sampling and Basic Plotting Outline Review of Probability Sampling A Special PMF Basic Plotting in R The Lab Assignment 2 I Go to previous lab http www stat psu edu mga 401 course info b lab Ch1 pdf I Go to next lab http www stat psu edu mga 401 course info b lab Ch3 pdf I Go to the Stat 401 home page http www stat psu edu mga 401 course info I http www stat psu edu mga http www google com I M George Akritas Lab2 Probability Sampling and Basic Plotting