Let X1, X2, …, Xn be independent and identically distributed...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent and identically distributed exponential random variables. Show that the probability that the largest of them is greater than the sum of the others is $n / 2^{n-1}$. That is, if $$ M=\max _{j} X_{j} $$ then show $$ P\left\{M>\sum_{i=1}^{n} X_{i}-M\right\}=\frac{n}{2^{n-1}} $$ Hint: What is $P\left[X_{1}>\sum_{i=2}^{n} X_{i}\right\} ?$

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent and identically distributed exponential random variables. Show that the probability that the largest of them is greater than the sum of the others is $n / 2^{n-1}$. That is, if
$$
M=\max _{j} X_{j}
$$
then show
$$
P\left\{M>\sum_{i=1}^{n} X_{i}-M\right\}=\frac{n}{2^{n-1}}
$$
Hint: What is $P\left[X_{1}>\sum_{i=2}^{n} X_{i}\right\} ?$

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution characterized by a constant hazard rate, which means that the probability of an event occurring in the next interval is independent of how much time has already elapsed. Its probability density function and cumulative distribution function have simple closed forms, and the memoryless property makes it especially useful in modeling the time between independent events, such as in queueing theory and reliability analysis.

Order Statistics

Order statistics deal with the properties of the sorted values of a sample of random variables. In the context of this problem, the maximum of a set of random variables is an order statistic, and understanding its behavior in relation to the other variables (such as being larger than the sum of the others) is crucial. Order statistics provide tools to analyze extremal events and are essential in statistical inference and risk analysis.

Memoryless Property

The memoryless property is a defining characteristic of the exponential distribution, which states that the probability of an event occurring in the future is independent of the past. This property simplifies various probability calculations, particularly when dealing with events that happen sequentially or conditionally, and is instrumental in deriving results involving waiting times or sums of exponential variables.

Convolution and the Gamma Distribution

When independent exponential random variables are summed, the resulting distribution is a gamma distribution. This process of summing independent random variables, known as convolution, is key in probabilistic analysis since it allows one to derive the distribution of aggregate measures. Understanding the relationship between exponential and gamma distributions is important in solving problems that involve the sum of waiting times or durations.

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