The waiting time, in hours, between successive speeders...

The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function $$ F(x)=\left\{\begin{array}{ll} 0_{+} & x<0, \\ 1-e^{-k x}, & x \geq 0. \end{array}\right. $$ Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of $X$; (b) using the probability density function of $X$.

The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function
$$
F(x)=\left\{\begin{array}{ll}
0_{+} & x<0, \\
1-e^{-k x}, & x \geq 0.
\end{array}\right.
$$
Find the probability of waiting less than 12 minutes between successive speeders
(a) using the cumulative distribution function of $X$;
(b) using the probability density function of $X$.

Key Concepts

Exponential Distribution

The exponential distribution is used to model the time between events in a Poisson process. It is characterized by a constant hazard rate and has memoryless properties, meaning that the probability of an event occurring in the future is independent of the past. In this context, it models the waiting time between successive speeders.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes a value less than or equal to a given number. For the exponential distribution, the CDF is typically expressed as F(x) = 1 - e^(-kx) for x ? 0, indicating the accumulated probability up to time x. This is directly applied to obtain the probability of an event occurring before a specified time.

Probability Density Function

The probability density function (PDF) represents the relative likelihood for the random variable to take on a particular value. For the exponential distribution, the PDF is derived as the derivative of the CDF and is given by f(x) = ke^(-kx) for x ? 0. It is used to compute probabilities over intervals by integrating over those intervals.

Unit Conversion

Unit conversion is important in probability problems, particularly when the units of measurement for time or other variables differ from those used in the provided distribution. In this case, converting minutes to hours is necessary to align with the time unit in the exponential distribution's function.

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