The steps in this exercise lead to the probability density...

The steps in this exercise lead to the probability density function of an Erlang random variable $X$ with parameters $\lambda$ and $r, f(x)=\lambda^{r} x^{r-1} e^{-\lambda x} /(r-1) !, x>0,$ and $r=1,2, \ldots$ (a) Use the Poisson distribution to express $P(X>x)$. (b) Use the result from part (a) to determine the cumulative distribution function of $X$. (c) Differentiate the cumulative distribution function in part (b) and simplify to obtain the probability density function of $X$.

The steps in this exercise lead to the probability density function of an Erlang random variable $X$ with parameters
$\lambda$ and $r, f(x)=\lambda^{r} x^{r-1} e^{-\lambda x} /(r-1) !, x>0,$ and $r=1,2, \ldots$
(a) Use the Poisson distribution to express $P(X>x)$.
(b) Use the result from part (a) to determine the cumulative distribution function of $X$.
(c) Differentiate the cumulative distribution function in part (b) and simplify to obtain the probability density function of $X$.

Key Concepts

Differentiation and the Probability Density Function

Differentiating the cumulative distribution function with respect to x yields the probability density function of the continuous random variable. For the Erlang distribution, this process involves applying differentiation techniques to the series expression arising from the Poisson probabilities, leading exactly to the known form of the Erlang probability density function.

Cumulative Distribution Function

The cumulative distribution function (CDF) for a continuous random variable gives the probability that the variable takes on a value less than or equal to a specific value. In the context of the Erlang distribution, the CDF can be derived using the survival function, which itself is expressed in terms of the Poisson probabilities corresponding to fewer than r events occurring by time x.

Poisson Distribution

The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time or space, assuming the events occur with a constant mean rate and independently of the time since the last event. It is used to relate the probability of no events having occurred within a time period to the survival function of the waiting time distribution in a Poisson process.

Erlang Distribution

The Erlang distribution is a special case of the gamma distribution that arises in the context of waiting times for events in a Poisson process. It is characterized by two parameters: a rate parameter and an integer shape parameter, which represents the number of events. Its probability density function expresses the likelihood of the waiting time until the r-th event occurs.

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