Understanding Discrete Random Variables in Statistics

Stat 321 – Lecture 10 Discrete random variables Did you hear about the guy whose hair was on fire and his feet were frostbitten? On average he was fine!

Announcements • Exam Thursday • Review sheet on web • Review problems and solutions on web • Covering chapters 1, 2; HW 1-3; Lab 1-3; Quiz 1-2 • You will be supplied with formulas (sample online) • You will be allowed to bring in one page of notes • Bring your calculator • Potential Review Q&A Tuesday

Example 1: Random Babies (cont.) • Number of “matches” S={ 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321} P(at most one match) ….

Random Variable • Let X represent the number of matches • Random Variable • assigns a number to each outcome in the sample space S={ 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321}

Random Variable • Let X = number of matches • Random Variable • assigns a number to each outcome in the sample space x={ 4 2 2 1 1 2 2 0 1 0 0 1 1 0 2 1 0 0 0 1 1 2 0 0}

Random Variable • Possible values of X: 0, 1, 2, 4 • Discrete R.V. • finite or countably infinite number of possible outcomes • Examples: • Number of heads in 5 tosses, X=0,1,2,3,4,5 • Number of tosses until first head X=1, 2, 3, … • Counter example: • Let X =time, height (“continuous”)

Probability Distribution for X • List of outcomes and their probabilities P(X=0)=9/24 = .375 P(X=1)=8/24 = .333 P(X=2)=6/24=.250 P(X=4)=1/24=.0417 • All probabilities are between 0 and 1 • Probabilities must sum to one

Probability distribution for X P(X=x) x 0 1 2 3 4

Expected value of X • E(X) = SxP(X=x) • Long-run average…

Cumulative distribution function • P(X < x) = S p(y) for y<x • F(2) = 23/24 = .9583 • F(-1) = P(X < -1) = 0 • F(5) = P(X < 5) = 1 • F(x) = P(X<x) for all values of x

Cumulative distribution function P(X<x) x 0 1 2 3 4

Example 2 • p(x) = 0 for x < 1 • p(1) = .3 • p(3) = .1 • p(4) = .05 • p(6) = .15 • p(12) = .40 • Makes sense? • Sums to one

Example 3 • p(2) = P(X=2) • P(X < 2) – P(X < 1) = .39-.19 = .20 • p(2.5) = P(X = 2.5) = 0 • F(2.5) = P(X < 2.5) = P(X < 2) = .39 • F(3) = P(X < 3) = .67 • P(X > 3) = 1-P(X<3) = 1-F(3) = 1-.67 = .33 • P(2 < X < 5) = F(5) – F(1) = .97-.19 = .78 • P(2 < X < 5) = F(4) – F(2) = .92 - .39 = .53

Example 4

Example 5 • Average daily max temperature Mean = 22.13 SD = 3.32 Mean = 71.84 SD = 5.98

For Tuesday • Bring questions on Ch. 1 and 2! • HW 3 due

Understanding Discrete Random Variables in Statistics