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Random Variables

Random Variables. Probability Continued Chapter 6. Random Variables.

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Random Variables

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  1. Random Variables Probability Continued Chapter 6

  2. Random Variables • Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution?

  3. Random Variable Example X = number of people who purchase electric hot tub X 0 1 2 3 .288 P(X) .216 .432 .064 (.6)(.6)(.6) GGG EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.4)(.4)

  4. Random Variables • A numerical variable whose value depends on the outcome of a chance experiment is called a random variable. • discrete versus continuous

  5. Discrete vs. Continuous • The number of desks in a classroom. • The fuel efficiency (mpg) of an automobile. • The distance that a person throws a baseball. • The number of questions asked during a statistics final exam.

  6. Discrete versus Continuous Probability Distributions Which is which? • Properties: • For every possible x value, 0 < x < 1. • Sum of all possible probabilities add to 1. • Properties: • Often represented by a graph or function. • Area of domain is 1.

  7. Means and Variances • The mean value of a random variable X (written mx ) describes where the probability distribution of X is centered. • We often find the mean is not a possible value of X, so it can also be referred to as the “expected value.” • The standard deviation of a random variable X (written sx )describes variability in the probability distribution.

  8. Mean of a Random Variable Example • Below is a distribution for number of visits to a dentist in one year. X = # of visits to the dentist. • Determine the expected value, variance and standard deviation.

  9. Formulas Mean of a Random Variable Variance of a Random Variable

  10. Mean of a Random Variable Example E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05) = 1.75 visits to the dentist

  11. Variance and Standard Deviation of a Random Variable Example Var(X) = (0 – 1.75)2(.1) + (1 – 1.75)2(.3) + (2 – 1.75)2(.4) + (3 – 1.75)2(.15) + (4 – 1.75)2(.05) = .9875

  12. Developing Transformation Rules • Consider the following distribution for the random variable X:

  13. X+1 • What is the probability distribution for X+1?

  14. 2X • What is the probability distribution for 2X?

  15. Consider • Suppose that E(X) = 2.5, Var(X) = 0.2 • What is E(X+5) = ?, Var(X+5) = ? • E(X+5) = 2.5 + 5 = 7.5 • Var(X+5) = 0.2 (no change) • What is E(X – 2.2) = ?, Var(X – 2.2) = ? • E(X – 2.2) = 2.5 – 2.2 = 0.3 • Var(X – 2.2) = 0.2 (no change)

  16. Consider • Suppose that E(X) = 2.5, Var(X) = 0.2 • What is E(3X) = ?, Var(3X) = ? • E(3X) = 3*2.5 = 7.5 • Var(3X) = 32 * 0.2 = 1.8 • What is E(2X – 1) = ?, Var(2X – 1) = ? • E(2X – 1) = 2(2.5) – 1 = 4 • Var(2X – 1) = 22 * 0.2 = 0.8

  17. Rule 1 • Rule 1: If X is a random variable and a and b are fixed numbers, then ma + bX = a + bmX • Rule 1: If X is a random variable and a and b are fixed numbers, then s2a + bX =b2s2X

  18. 1 .2 1 .2 .8 2 .2 1 .8 2 .8 2 X+X • What is the probability distribution for X+X?

  19. X+X

  20. 1 .2 1 .2 .8 2 .2 1 .8 2 .8 2 X–X • What is the probability distribution for X–X ?

  21. X – X

  22. Consider • Suppose that E(X) = 2.5, Var(X) = .16, E(Y) = 1.2, Var(Y) = .36 • What is E(X+Y) = ?, Var(X+Y) = ? • E(X+Y) = 2.5 + 1.2 = 3.7 • Var(X+Y) = .16 + .36 = .52 • E(X – Y) = ?, Var(X – Y) = ? • E(X – Y) = 2.5 – 1.2 = 1.3 • Var(X – Y) = .16 + .36 = .52

  23. Consider • E(X) = 2.5, Var(X) = .16, E(Y) = 1.2, Var(Y) = .36 • What is s(X) = ?, s(Y) = ?, s(X+Y) = ? • s(X) = .4, s(Y) = .6 • Since Var(X+Y) = .52, s(X+Y) = .7211 • Cannot add standard deviation directly!! • What is E(2X – 4Y) = ?, s(2X-4Y) = ? • E(2X–4Y) = 2(2.5) – 4(1.2) = 0.2 • Var(2X–4Y) = 22(.16) + 42(.36) = 6.4 • So s(2X–4Y) = 2.53

  24. Rule 2 • Rule 2: If X and Y are random variables, then mX + Y = mX + mY mX – Y = mX – mY • Rule 2: If X and Y are independent random variables, then s2X + Y = s2X + s2Y s2X - Y = s2X + s2Y • Note: cannot combine standard deviation directly.

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