Consider the following exponential probability density...

Consider the following exponential probability density function.
\[
f(x)=\frac{1}{3} e^{-x / 3} \quad \text { for } x \geq 0
\]
a. Write the formula for $P\left(x \leq x_{0}\right)$
b. Find $P(x \leq 2)$
c. Find $P(x \geq 3)$
d. Find $P(x \leq 5)$
e. Find $P(2 \leq x \leq 5)$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time or space between events in a Poisson process. It is defined by its probability density function (pdf), which typically takes the form f(x) = (1/?) e^(-x/?) for x >= 0, where ? > 0 is the scale parameter. This distribution has properties like the memoryless property, meaning that future probabilities do not depend on past outcomes.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a continuous random variable specifies the probability that the variable takes a value less than or equal to a given x. For the exponential distribution, the CDF is derived by integrating the pdf from 0 to x, resulting in F(x) = 1 - e^(-x/?) for x >= 0. The CDF is essential in calculating probabilities over intervals for continuous distributions.

Integration for Probability Calculations

For continuous probability distributions, computing the probability that a variable falls within a specific interval involves integrating the probability density function (pdf) over that interval. This technique is used to move from the pdf to cumulative probabilities and to compute probabilities for ranges, such as P(a <= x <= b), by calculating the difference F(b) - F(a) where F(x) is the CDF.

Transcript

Please give Ace some feedback