Cumulative distribution function (CDF)

Definition

A cumulative distribution function (CDF) describes the probability that a continuous random variable takes on a value less than or equal to a given point. It is a non-decreasing function that ranges from 0 to 1.

5 Must Know Facts For Your Next Test

1. The CDF of an exponential distribution with rate parameter $\lambda$ is $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$.
2. A CDF is always non-decreasing and right-continuous.
3. The value of the CDF at negative infinity is 0, and at positive infinity it is 1.
4. To find the probability that a variable falls between two values, subtract the CDF values at those points: $P(a < X \leq b) = F(b) - F(a)$.
5. For an exponential distribution, the mean ($\frac{1}{\lambda}$) and standard deviation are both equal to $\frac{1}{\lambda}$.

Review Questions

• What is the formula for the CDF of an exponential distribution with rate parameter $\lambda$?
• How can you use the CDF to find the probability that a variable falls between two values?
• What are the range and key properties of a CDF?

Related terms

Probability Density Function (PDF):

A function that describes the likelihood of a continuous random variable taking on a particular value. The PDF for an exponential distribution with rate parameter $\lambda$ is $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$.

Rate Parameter ($\lambda$): A constant used in defining exponential distributions, representing the rate at which events occur. It is the inverse of the mean.

$e^{-x}$: $e^{-x}$ represents the exponential decay function, commonly appearing in formulas related to exponential distributions. It describes how quantities decrease over time at a constant proportional rate.