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Understanding Probability Distributions and Random Variables in Mathematical Examples

Learn about defining probabilities, random variables, and examples in mathematical contexts. Explore discrete and continuous probability distributions to solve various scenarios and calculations.

quail-warner
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Understanding Probability Distributions and Random Variables in Mathematical Examples

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  1. Defining Probabilities: Random Variables • Examples: • Out of 100 heart catheterization procedures performed at a local hospital each year, the probability that more than five of them will result in complications is __________ • Drywall anchors are sold in packs of 50 at the local hardware store. The probability that no more than 3 will be defective is __________ • In general, ___________

  2. Discrete Random Variables • Example: • Look back at problem 3, page 46. Assume someone spends $75 to buy 3 envelopes. The sample space describing the presence of $10 bills (H) vs bills that are not $10 (N) is: _____________________________ • The random variable associated with this situation, X, reflects the outcome of the choice and can take on the values: _____________________________

  3. Discrete Probability Distributions • The probability that there are no $10 in the group is P(X = 0) = ___________________ (recall results from last time) • The probability distribution associated with the number of $10 bills is given by:

  4. Another Example • Example 3.3, pg 66 P(X = 0) = _____________________

  5. Discrete Probability Distributions • The discrete probability distribution function (pdf) • f(x) = P(X = x) ≥ 0 • Σxf(x) = 1 • The cumulative distribution,F(x) • F(x) = P(X ≤ x) = Σt ≤ xf(t)

  6. Probability Distributions • From our example, the probability that no more than 2 of the envelopes contain $10 bills is P(X ≤ 2) = F(2) = _________________ • The probability that no fewer than 2 envelopes contain $10 bills is P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - F(1) = ________________

  7. Another View • The probability histogram

  8. Your Turn … • The output from of the same type of circuit board from two assembly lines is mixed into one storage tray. In a tray of 10 circuit boards, 6 are from line A and 4 from line B. If the inspector chooses 2 boards from the tray, show the probability distribution function associated with the selected boards being from line A.

  9. Continuous Probability Distributions • Examples: • The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is __________ • The probability that a given part will fail before 1000 hours of use is __________ • In general, __________

  10. Homework due dates • Friday, 9/3 pg.54-56 • Wednesday, 9/8 pg. 72-74 (both sets) (Check the web site on Thursday, 9/2, for an updated schedule.)

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