A spinner from a board game randomly indicates a real number...

A spinner from a board game randomly indicates a real number between 0 and $10 .$ The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length. (a) Explain why the function $f(x)=\left\{\begin{array}{ll}{0.1} & {\text { if } 0 \leqslant x \leqslant 10} \\ {0} & {\text { if } x<0 \text { or } x>10}\end{array}\right.$ is a probability density function for the spinner's values. (b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

A spinner from a board game randomly indicates a real number between 0 and $10 .$ The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.
(a) Explain why the function
$f(x)=\left\{\begin{array}{ll}{0.1} & {\text { if } 0 \leqslant x \leqslant 10} \\ {0} & {\text { if } x<0 \text { or } x>10}\end{array}\right.$
is a probability density function for the spinner's values.
(b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

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Key Concepts

Probability Density Function (PDF)

A probability density function is a function that describes the relative likelihood for a continuous random variable to take on a given value. It integrates over an interval to give the probability that the variable falls within that interval, and it must be non-negative everywhere with the total integral equal to 1.

Uniform Distribution

A uniform distribution is a type of probability distribution in which all outcomes in a specified range are equally likely. This means that the probability density function is constant across the interval of interest, reflecting that no subinterval has a higher or lower chance of being selected than any other of the same length.

Expected Value

The expected value of a continuous random variable is essentially the 'mean' or average outcome, computed as the integral of the product of the variable and its probability density function over the possible values. It provides a measure of the central tendency of the distribution.

Integration in Probability Theory

Integration is a key tool in probability theory used to compute probabilities, expected values, and other statistical measures when dealing with continuous random variables. It allows one to sum up the contributions from infinitesimally small intervals across the entire range of the variable.

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