Find the distribution of the number of observations until the value of the first observation is exceeded.

Understanding
The \(X_i\) shown below are independent and identically distributed (iid) observations. The following equation is the probability that the \(n\)th observation is the first to exceed \(X_1\) given that \(X_1=x\). \[P(N=n\mid X_1=x)=P(X_2\le x)\cdots P(X_{n-1}\le x)P(X_n>x)\] The distribution that we seek is \(P(N=n)\) hence the conditional part of the probability has to be integrated out. The function \(f(x)\) is the probability density function (pdf) of the observations. Then by total probability \[P(N=n)=\int_{-\infty}^\infty P(N=n\mid X_1=x)f(x)\,dx\] In the following method, \(F(x)\) is the cumulative distribution function (cdf) of the observations. It follows that \(F(x)=P(X_i\le x)\). For a cdf, \(F(-\infty)=0\), \(F(\infty)=1\), and \(dF(x)=f(x)\,dx\).

Method

 * 1) Perform the first step, as shown below:\[\begin{align*}P(N=n)&=\int_{-\infty}^\infty P(N=n\mid X_1=x)f(x)\,dx\\&=\int_{-\infty}^\infty P(X_2\le x)\cdots P(X_{n-1}\le x)P(X_n>x)f(x)\,dx\\&=\int_{-\infty}^\infty F(x)^{n-2}(1-F(x))f(x)\,dx\end{align*}\]
 * 2) Use \(t=F(x)\) and \(dt=dF(x)=f(x)\,dx\) as a substitution and insert it into the previous expression, as shown below:\[\begin{align*}P(N=n)&=\int_0^1t^{n-2}(1-t)\,dt\\&=\left.\left(\frac{t^{n-1}}{n-1}-\frac{t^n}{n}\right)\right|_0^1\\&=\frac{1^{n-1}}{n-1}-\frac{1^n}{n}-\frac{0^{n-1}}{n-1}+\frac{0^n}{n}\\&=\frac{1}{n-1}-\frac{1}{n}\\&=\frac{n}{n(n-1)}-\frac{n-1}{n(n-1)}\\&=\frac{1}{n(n-1)}\end{align*}\]

Everything else
NB: The method section still needs to be completed. It's written as two large mathematical facts rather than a sequence of verbal commands to follow in order to perform the manipulations of each step of working.

The probability that \(X_2>X_1\) is \(\frac{1}{2}\). The probability that \(X_2\le X_1\) and \(X_3>X_1\) is \(\frac{1}{6}\). The interesting thing is that all of this is independent of \(f(x)\), the distribution of the individual observations.

Reference
Casella and Berger, Statistical Inference, Second Edition, p. 255.

Navigation

 * Statistics
 * Mathematics
 * Topics (listed alphabetically)
 * Main Page