Question

Let \(X_{1}, X_{2}, \ldots\) be independent and identically distributed nonnegative continuous random variables having density function \(f(x) .\) We say that a record occurs at time \(n\) if \(X_{n}\) is larger than each of the previous values \(X_{1}, \ldots, X_{n-1} .\) (A record automatically occurs at time 1.) If a record occurs at time \(n\), then \(X_{n}\) is called a record value. In other words, a record occurs whenever a new high is reached, and that new high is called the record value. Let \(N(t)\) denote the number of record values that are less than or equal to \(t .\) Characterize the process \(\\{N(t), t \geqslant 0\\}\) when (a) \(f\) is an arbitrary continuous density function. (b) \(f(x)=\lambda e^{-\lambda x}\). Hint: Finish the following sentence: There will be a record whose value is between \(t\) and \(t+d t\) if the first \(X_{i}\) that is greater than \(t\) lies between \(\ldots\)

Short answer

In conclusion, the probability of a record occurrence at any time \(t\) for a given continuous density function \(f(x)\) can be represented by differentiating the expression \(\mu_n\), as \(P[\text{record between } t \text{ and } t + dt] = \frac{d\mu_n}{dt} dt\). Specifically for the exponential density function, the probability of a record occurrence between \(t\) and \(t+dt\) can be computed using \(\mu_n = \int_0^t x[\lambda e^{-\lambda x}]^n \lambda e^{-\lambda x} dx\).

Step-by-step solution

Understand records and record values

A record occurs at time \(n\) if the random variable \(X_n\) is larger than each of the previous values \(X_1, X_2, ... X_{n-1}\). A record value is the value of \(X_n\) when a record occurs. Step 2: Find the probability of a record occurrence.

Probability of record occurrence

To find the probability of a record occurrence at time \(n\), we need to find the probability that \(X_n\) is greater than \(X_1, X_2, ... X_{n-1}\). As \(X_1, X_2, ...\) are independent and identically distributed, the probability of a record occurring at time \(n\) is given by \(\frac{1}{n}\). Now we will characterize the process \(N(t)\) for the given density functions. Step 3: Find the probability of a record occurrence at any time \(t\) for a given \(f\).

General density function case

For a general continuous density function \(f(x)\), let's find the probability of a record occurrence between \(t\) and \(t+dt\). Since the previous record value must be less than \(t\) and the first value greater than \(t\) occurs between \(t\) and \(t+dt\), we can write the probability as: \[\mu_n = \int_0^t x (f(x))^n f(x) \, dx\] Now, differentiate this expression with respect to \(t\) to get the probability of a record occurrence in the limit as \(dt \rightarrow 0\). Step 4: Differentiate \(\mu_n\)

Differentiate the expression

We can express the probability of a record occurrence between \(t\) and \(t+dt\) as: \[P[\text{record between } t \text{ and } t + dt] = \frac{d\mu_n}{dt} dt\] Now, we can use the given density function \(f(x) = \lambda e^{-\lambda x}\) and plug it in the expression. Step 5: Characterize the process \(N(t)\) for the given density function.

Characterize \(N(t)\) for exponential density function

For the density function \(f(x) = \lambda e^{-\lambda x}\), the probability of a record occurrence between \(t\) and \(t+dt\) will be: \[\mu_n = \int_0^t x[f(x)]^n f(x) dx = \int_0^t x[\lambda e^{-\lambda x}]^n \lambda e^{-\lambda x} dx\] Now, differentiate this expression to find the probability of a record occurrence in the limit as \(dt \rightarrow 0\): \[P[\text{record between } t \text{ and } t + dt] = \frac{d\mu_n}{dt} dt\] Concluding with these results, we have characterized the process \(N(t)\) for arbitrary continuous density functions (the general case) and for the exponential density function.