Cumulative Distribution Function

Sampling

Probability Density Function

Reliability Function

Hazard Function

Expectation

Exam

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The cumulative distribution function, F(x), denotes the area beneath the probability density function to the left of x. This is demonstrated in the figure below.

Cumulative Distribution Function

The area of the shaded region of the probability density function in the figure is 0.2525. This is the corresponding value of the cumulative distribution function at x = 190.  Mathematically, the cumulative distribution function is equal to the integral of the probability density function to the left of x.

Example:  A random variable has the probability density function f(x) = 0.125x, where x is valid from 0 to 4. What is the probability of x being less than or equal to 2?

Solution:

Example:  The time to fail for a transistor has the following probability density function. What is the probability of failure before t = 200?

t > 0

Solution:

The probability of failure before t = 200 is

The integral of f(t) is

-exp(-0.01t)

evaluating this expression from 0 to 200 is

P(t<200) = -exp[-0.01(200)] - {-exp[-0.01(0)]} = -0.135 - (1) = 0.865