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Understanding Discrete Random Variables: Binomial Distribution Analysis

Explore the concept of discrete random variables, probability mass function, expected value, variance, and functions involving random variables. Practice with examples and pop quizzes for a comprehensive understanding.

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Understanding Discrete Random Variables: Binomial Distribution Analysis

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  1. Stat 321 – Lecture 12 Discrete Random Variables (cont.) Binomial Probability Distribution (2.4)

  2. Last Time – Discrete RVs • Random variable = maps every outcome in sample space to a numerical value • Discrete = can “list” every possible value • Probability mass function (pmf) • List every outcome, assign probability • Line graph/probability histogram • Expected value E(X) = Sxi P(X=xi) • Long-run average • Cumulative distribution function • F(x) = P(X <x) • P(X > x) = 1-F(x) • P(a < X < b) = F(b) –F(a-)

  3. Other properties • Variance • Interpretation • Short-cut formula: V(X) = E(X2) – [E(X)]2

  4. E(Y) = 3(.15)+ 6(.35) + 9(.5) = 7.05 V(Y)=(3-7.05)2(.15)+(6-7.05)2(.35)+(9-7.05)2(.5) = 4.75 SD(Y) = 2.18 E(X) = 1(.3)+3(.1)+4(.05)+6(.15)+12(.4) = 6.5 E(X2)=1(.3)+9(.1)+16(.05)+36(.15)+144(.4)= 65 V(X) = 65 – (6.52) = 22.75 SD(X) = 4.17

  5. Functions of random variables • In general, E(h(X)) = Sh(x)P(X=x) • E.g., E(X2) • But what about h(X) = aX + b? • E.g., h(X) = 5/9X – 160/9 • Cut in half and subtract 18 • SLO temps, mean  72, std dev  6 • Cut in half and subtract 18? 22, -15?? • When have a linear function • E(aX+b) = aE(X)+b, SD(aX+b) = |a|SD(X)

  6. Pop Quiz • Take out a blank piece of paper • Individual • Multiple choice, 3 options (A, B, C) • 10 questions

  7. Exchange with Neighbor • Record number correct

  8. “Solutions” • C • A • B • C • C • C • B • A • C • C

  9. Example 1 • Probability Distribution • X = number correct out of 4 multiple choice questions (3 options each) • Possible values for X? • Probability for each value? • What assumptions are you making?

  10. Example 1 x 0 1 2 3 4 p(x) .20 .40 .30 .10 .01 P(X>2) = .3+.1+.01=.41 P(X>2) = 1-P(X<1) = 1-.2-.4=.4

  11. Example 1 • General form • Conditions • Two possible outcomes • Constant probability of success, p • Trials are independent • Random variable counts the number of successes in a fixed number of trials, n

  12. Example 1 • What if there are 10 questions? • How find P(passing) • P(X> 5) = 1-P(X< 4) • Table • Minitab

  13. Example 1

  14. For Tuesday • HW 4… • Exam 1 Mean = .82, Median = .84 Solutions in Blackboard Course Avg updated Correct answers vs. highest scoring answers/ justification

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