Let X be a continuous random variable with density function...

Let $X$ be a continuous random variable with density function $$f(x)=\left\{\begin{array}{cc}{\frac{|x|}{10}} & {\text { for }-2 \leq x \leq 4} \\ {0} & {\text { otherwise }}\end{array}\right.$$ Calculate the expected value of $X .$ Choose one of the following. Source: Society of Actuaries. $\begin{array}{ll}{\text { (a) } 1 / 5} & {\text { (b) } 3 / 5} \\ {\text { (c) } 1} & {\text { (d) } 28 / 15} \\ {\text { (e) } 12 / 5}\end{array}$

Let $X$ be a continuous random variable with density function
$$f(x)=\left\{\begin{array}{cc}{\frac{|x|}{10}} & {\text { for }-2 \leq x \leq 4} \\ {0} & {\text { otherwise }}\end{array}\right.$$
Calculate the expected value of $X .$ Choose one of the following. Source: Society of Actuaries.
$\begin{array}{ll}{\text { (a) } 1 / 5} & {\text { (b) } 3 / 5} \\ {\text { (c) } 1} & {\text { (d) } 28 / 15} \\ {\text { (e) } 12 / 5}\end{array}$

Key Concepts

Piecewise Function

A piecewise function is defined by different expressions over different parts of its domain. This approach is often used in probability to specify a pdf that behaves differently on separate intervals, ensuring that the overall function satisfies the properties required of a valid probability density function.

Integration

Integration is the mathematical process used to accumulate the contributions of continuously distributed probabilities, such as computing the expected value. In probability, it serves to sum or average values across an interval, weighted by the density function, which is essential in continuous distributions.

Continuous Random Variable

A continuous random variable is one that can take any value in an interval and is described by a probability density function. This type of variable is fundamental in probability theory and statistics, where probabilities are determined over intervals rather than individual points.

Probability Density Function

A probability density function (pdf) defines the likelihood of a continuous random variable taking on a specific value. The function is non-negative and its integral over the entire space is equal to 1, ensuring it properly represents a probability distribution.

Expected Value

The expected value of a continuous random variable, often called the mean, is a measure of the central tendency of the distribution. It is computed as the integral of x multiplied by its probability density function over the variable's entire range, which provides a weighted average of all possible values.

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