Annual Rainfall The annual rainfall in a remote Middle...

Annual Rainfall The annual rainfall in a remote Middle Fastern country varies from 0 to 5 in. and is a random variable within probability density function defined by $f(x)=\frac{5.5-x}{15}$ Find the following probabilities for the annual rainfall in a randomly selected year. (a) The probability that the annual rainfall is greater than 3 in. (b) The probability that the annual rainfall is less than 2 in. (c) The probability that the annual rainfall is between 1 in. and 4 in.

Annual Rainfall The annual rainfall in a remote Middle Fastern country varies from 0 to 5 in. and is a random variable within probability density function defined by
$f(x)=\frac{5.5-x}{15}$
Find the following probabilities for the annual rainfall in a randomly selected year.
(a) The probability that the annual rainfall is greater than 3 in.
(b) The probability that the annual rainfall is less than 2 in.
(c) The probability that the annual rainfall is between 1 in. and 4 in.

Key Concepts

Definite Integration in Probability

Definite integration is the mathematical process used to calculate the area under the curve of the probability density function over a specific interval. This area represents the probability that the random variable falls within that interval. It is essential in converting the abstract function into concrete probability values.

Cumulative Distribution Function (CDF)

The cumulative distribution function gives the probability that a continuous random variable is less than or equal to a certain value. It is obtained by integrating the probability density function from the lower limit of the variable's domain up to the value of interest, and it is used to calculate the probability over a range by taking differences of CDF values.

Continuous Random Variable

A continuous random variable is one that can assume any value within a given range or interval. Since the number of possible outcomes is uncountably infinite, probabilities are defined over intervals rather than at specific points, and techniques from calculus, such as integration, are used to compute these probabilities.

Probability Density Function (PDF)

A probability density function describes the likelihood of a continuous random variable taking on a particular value. The function must be nonnegative and integrate to one over its entire domain. It serves as the fundamental tool for determining probabilities by integrating over intervals, representing the area under the curve of the function.

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