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Continuous Random Variables

When the world gets continuous, calculus meets probability.

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A continuous random variable is a random variable that is used to model a situation where there are uncountably many outcomes.

For example, the amount of time it takes for a runner to finish a race would be modeled by a continuous random variable since the time can be any positive real number (assuming you can measure time with arbitrary precision).

The probability density function of a continuous random variable gives the relative likelihood of any outcome in a continuum occurring. The probability that a random variable \(X\) takes a value in the (open or closed) interval \([a,b]\) is given by the integral of a function called the probability density function \(f_X(x)\):

\[P(a\leq X \leq b) = \int_a^b f_X(x) \,dx.\]

In this sense, the probability density function is not the "probability" of \(x\) occurring, but rather a measure of relative likelihood that can be used in the equation above to determine how often the continuous random variable lies in some interval.

At any time, the time until the next call is received by a call center is given by a random variable \(X\) with the probability density function \(f_X(x) = 5 e^{-5x},\) where \(x\in [0,\infty)\) denotes the time in hours passed from the given moment until the next call. At some time, what is the expected time to wait until the next call?

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