ch16 - STA 2023 - Holbrook Random Variables Chapter 16 16 -...

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Unformatted text preview: STA 2023 - Holbrook Random Variables Chapter 16 16 - 1 Data Types STA 2023 - Holbrook D a ta Q u a n t ita t iv e D is c r e t e 16 - 2 C a t e g o r ic a l C o n t in u o u s Data Types STA 2023 - Holbrook D a ta Q u a n t ita t iv e D is c r e t e 16 - 3 C a t e g o r ic a l C o n t in u o u s STA 2023 - Holbrook Discrete Random Variables 16 - 4 Discrete Random Variable STA 2023 - Holbrook 1. Random Variable Assumes a numerical value for each Assumes outcome of an Experiment outcome Example: Number of Tails in 2 Coin Tosses 2. Discrete Random Variable 2. 16 - 5 Whole Number (0, 1, 2, 3 etc.) Obtained by Counting Usually Finite Number of Values Discrete Random Variable Examples STA 2023 - Holbrook Experiment Random Variable Make 100 Sales Calls # Sales Inspect 70 Radios Possible Values 0, 1, 2, ..., 100 # Defective 0, 1, 2, ..., 70 Answer 33 Questions # Correct 0, 1, 2, ..., 33 Count Cars at Toll # Cars Between 11:00 & 1:00 Arriving 0, 1, 2, ..., ∞ 0, 16 - 6 Data Types STA 2023 - Holbrook D a ta Q u a n t ita t iv e D is c r e t e 16 - 7 C a t e g o r ic a l C o n t in u o u s STA 2023 - Holbrook Continuous Random Variables 16 - 8 Continuous Random Variable STA 2023 - Holbrook 1. Random Variable Assumes a numerical value for each outcome of Assumes an Experiment an Weight of a Student (e.g., 115, 156.8, etc.) Weight 2. Continuous Random Variable Whole or Fractional Number Obtained by Measuring Infinite Number of Values in Interval 16 - 9 Too Many to List Like Discrete Variable Continuous Random Variable Examples STA 2023 - Holbrook Experiment Random Variable Possible Values Weigh 100 People Weight 45.1, 78, ... Measure Part Life Hours 900, 875.9, ... Ask Food Spending Spending 54.12, 42, ... Measure Time Between Arrivals Inter-Arrival 0, 1.3, 2.78, ... Time 16 - 10 10 STA 2023 - Holbrook Discrete Probability Distributions 16 - 11 11 Discrete Probability Models STA 2023 - Holbrook 1. List of All possible [x, p(x)] pairs x = Value of Random Variable (Outcome) p(x) = Probability Associated with Value 2. Mutually Exclusive (No Overlap) 3. Collectively Exhaustive (Nothing Left Out) 4. 0 ≤ p ( x ) ≤ 1 4. 5. Σ p ( x ) = 1 5. 16 - 12 12 Discrete Probability Model Example STA 2023 - Holbrook Experiment: Toss 2 Coins. Count # Tails. Experiment: Probability Distribution Values, x Probabilities, p(x) Values, 0 1 16 - 13 13 2/4 = .50 2 © 1984-1994 T/Maker Co. 1/4 = .25 1/4 = .25 1/4 Summary Measures STA 2023 - Holbrook 1. Expected Value Mean of Probability Distribution Weighted Average of All Possible Values µ = E(X) = Σx p(x) 2. Variance 16 - 14 14 Weighted Average Squared Deviation Weighted about Mean σ2 = E[ (x − µ) 2 ] = Σ (x − µ) 2 p(x) µ) µ) Thinking Challenge STA 2023 - Holbrook You toss 2 coins. You’re You interested in the number interested of tails. What are the expected value & expected standard deviation of standard of this random variable, number of tails? number © 1984-1994 T/Maker Co. 16 - 15 15 Expected Value & Variance Solution* STA 2023 - Holbrook 2 2 x p(x) x p(x ) x-µ (x -µ ) (x -µ ) 2 p( x ) p( 0 .25 0 -1.00 1.00 .25 1 .50 .50 0 0 0 2 .25 .50 1.00 1.00 .25 µ = 1.0 1.0 16 - 16 16 2 σ 2 = .50 .50 STA 2023 - Holbrook Skip: Pages 414­419 and 423­424 (Pages 307­310 First Edition) 16 - 17 17 End of Chapter Any blank slides that follow are blank intentionally. ...
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