Problem 45 - Calculus for Biology and Medicine

Question

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with uniform distribution on \((0,1)\). Define \(X=\min \left(X_{1}, X_{2}, \ldots,\right.\), \(\left.X_{n}\right)\) (a) Compute \(P(X>x)\). (b) Show that \(P(X>x / n) \rightarrow e^{-x}\) as \(n \rightarrow \infty\).

Step-by-Step Solution

Answer

(a) \( P(X > x) = (1-x)^n \). (b) \( P(X > x/n) \rightarrow e^{-x} \) as \( n \rightarrow \infty \).

Step 1: Understanding the problem

We want to compute the probability that the minimum of random variables is greater than a certain value and check the behavior of this probability under certain conditions in the limit.

Step 2: Probability of X>x

For each of the independent uniform random variables \( P(X_i > x) = 1 - x \) assuming \( X_i \) is uniformly distributed on \( (0,1) \). Therefore, the probability that all \( n \) variables are greater than \( x \) is \[P(X > x) = P(X_1 > x) P(X_2 > x) \ldots P(X_n > x) = (1-x)^n\] by the independence property.

Step 3: Limit behavior of P(X>x/n)

Now let's consider \(P\left(X > \frac{x}{n}\right) = \left(1 - \frac{x}{n}\right)^n\).As \( n \rightarrow \infty \), we can use the limit definition of the exponential function. We know \[\left(1 - \frac{x}{n}\right)^n \rightarrow e^{-x}\] from the standard limit \( \left(1 + \frac{a}{n}\right)^n \rightarrow e^a \) as \( n \rightarrow \infty \).

Key Concepts

Uniform Distribution

Uniform distribution is a special probability distribution where every outcome is equally likely. Imagine a perfectly balanced dice where every number has the same chance of landing face up. In mathematical terms, a uniform distribution over the interval \(0, 1\) means any point between 0 and 1 is just as likely to be chosen.
For example, if we have a variable \(X\) that is uniformly distributed over \(0,1\), the probability density function is constant across this interval. So, the probability of selecting a value greater than some \(x\) is simply \(1-x\), showing that as \(x\) increases, the probability decreases.

• Uniformly distributed variables are often used to model situations where every outcome is equally possible.
• They're crucial in understanding random selections.

Understanding uniform distribution helps in solving problems where each possible outcome within a certain range is equally likely, as in this exercise.

Random Variables

Random variables are fundamental in probability and statistics, representing outcomes of random phenomena. In more intuitive terms, consider them as containers that hold numbers determined through random processes. In our scenario, \(X_1, X_2, \ldots, X_n\) are random variables that can take any value between 0 and 1 due to their uniform distribution.
Each random variable is independent in our example. This independence is crucial because it allows us to consider the combined probability of a series of events by multiplying their individual probabilities.

• Random variables can be discrete or continuous. In our case, \(X_i\) are continuous over the interval \(0,1\).
• They describe real-world scenarios such as the likelihood that a system's component will fail within a certain time frame.

Grasping random variables is essential for any statistical problem as they form the backbone of probability models.

Limit Behavior

Limit behavior describes how a function behaves as the input approaches some value, often infinity. It is a key concept in understanding how changing parameters affect probability.
When studying our exercise, we focus on the probability \(P\left(X > \frac{x}{n}\right)\) as \(n\), the number of random variables, becomes very large. We observe the expression \(\left(1 - \frac{x}{n}\right)^n\). This is tied to a well-known limit, \(\left(1 + \frac{a}{n}\right)^n ightarrow e^a\) as \(n ightarrow \infty\).

• This showcases how a probability structure can simplify into an exponential function.
• It helps in determining long-run behavior of stochastic processes.

Understanding limit behavior aids in making sense of long-term trends in probabilistic scenarios and can simplify complex calculations.

Exponential Function

The exponential function is one of the most important mathematical functions, especially in growth and decay models. It's defined as \(e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. It grows rapidly and is key in describing natural phenomena.
In relation to our problem, we see how the probability \(P\left(X > \frac{x}{n}\right)\) tends towards \(e^{-x}\) using the limit. This exponential behavior describes a rapid decay in probability as \(x\) increases.

• The concept is widely used in diverse fields such as engineering, physics, and finance.
• Its occurrence in the problem demonstrates how exponential decay describes the decreasing likelihood of a random variable exceeding a threshold over infinitely many trials.

Learning about exponential functions enables deeper insight into processes where change accelerates at a rate proportional to the current value.