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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?

What proportion of patient recovery times (x) are less than 60 min?

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Probability Density Function

• A (probability) density function f(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.

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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.

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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.

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Calculating Probabilities from a Density Function

The following density

function shows the daily

water usage

(x in millions of gallons) in

Gotham City.

a. Find the probability the

daily water usage is between

1 million and 4 million gallons.

b. Find the probability that 3 million gallons will not be sufficient to meet daily water usage demand.