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Probability and Probability Density Functions

Probability and Probability Density Functions. A random variable x is a variable whose numerical value depends on chance. For example, What is the probability that a patient’s recovery time ( x ) is between 40 min and 50 min?

cathleen-leblanc
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Probability and Probability Density Functions

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  1. Probability and Probability Density Functions A random variablex is a variable whose numerical value depends on chance. For example, What is the probability that a patient’s recovery time (x) is between 40 min and 50 min? What proportion of patient recovery times (x) are less than 60 min?

  2. Probability Density Function • A (probability) density functionf(x) for a random variable x is a continuous function that satisfies three properties: (1) the function outputs must be >0, that is, the graph of the function cannot go below the x-axis; (2) the total area between the graph of the function and the horizontal axis is 1; (3) the probability values for x correspond to area values. That is P(a<x<b) = area between the graph and the x-axis that is bounded on the left by x=a and on the right by x=b.

  3. Example of a (Continuous) Density Function • A continuous function whose graph is never below the x-axis. • The total area between the graph and the x-axis is 1. • Probabilities are areas.

  4. Calculating Probabilities from a Density Function The following density function shows the proportion of patients who recover from an illness x minutes after receiving treatment. a. Find the probability that the recovery time is between 12 min and 36 min. b. Find the probability that the recovery time takes no more than 24 min.

  5. Calculating Probabilities from a Density Function The following density function shows the daily water usage (x in millions of gallons) in Gotham City. • Find the probability the daily water usage is between 1 million and 4 million gallons. • Find the probability that 3 million gallons will not be sufficient to meet daily water usage demand.

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