Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables, each having a uniform distribution over \((0,1)\). Let \(M=\) maximum \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\). Show that the distribution function of \(M, F_{M}(\cdot)\), is given by $$ F_{M}(x)=x^{n}, \quad 0 \leq x \leq 1 $$ What is the probability density function of \(M ?\)

Short answer

The probability density function of \(M\), the maximum of \(n\) independent random variables with a uniform distribution over (0,1), is given by: $$ f_M(x) = n \cdot x^{(n-1)}, \quad 0 \le x \le 1 $$

Step-by-step solution

Find the probability of \(M \le x\)

We want to find the probability that the maximum of the random variables \(X_1, X_2, \dots, X_n\) is less than or equal to \(x\), i.e., \(P(M \le x)\). Since the random variables are independent, we can find the product of their individual probabilities of being less than or equal to \(x\): $$ P(M \le x) = P(X_1 \le x) \cdot P(X_2 \le x) \cdots P(X_n \le x) $$

Compute individual probabilities

Since each random variable \(X_i\) has a uniform distribution over (0,1), their cumulative distribution functions (CDFs) can be written as: $$ P(X_i \le x) = \begin{cases} 0 & x < 0 \\ x & 0 \le x \le 1 \\ 1 & x > 1 \end{cases} $$

Compute the cumulative distribution function of \(M\), \(F_M(x)\)

Using the individual probabilities, we can now find \(F_M(x) = P(M \le x)\). Since the range of the variables is (0,1), we can ignore the cases when \(x < 0\) and \(x > 1\): $$ F_M(x) = P(M \le x) = P(X_1 \le x) \cdot P(X_2 \le x) \cdots P(X_n \le x) = x \cdot x \cdots x = x^n $$ So, the cumulative distribution function of \(M\) is \(F_M(x) = x^n\) for \(0 \le x \le 1\).

Find the probability density function of \(M\)

To find the probability density function (PDF) of \(M\), we need to differentiate the cumulative distribution function with respect to \(x\): $$ \frac{d}{dx} F_M(x) = \frac{d}{dx} x^n = n \cdot x^{(n-1)} $$ Thus, the probability density function of \(M\) is: $$ f_M(x) = n \cdot x^{(n-1)}, \quad 0 \le x \le 1 $$

Key concepts

Independent Random Variables

When we speak of independent random variables, we're referring to a set of random variables where the outcome of any one variable does not affect the outcome of the others. This is a critical concept in probability theory because it simplifies calculations and allows for the use of product rules. For example, if we toss two coins, the result of the first toss has no bearing on the result of the second toss; these are independent events.

Mathematically, two random variables, say, A and B, are independent if the probability of both happening simultaneously, P(A and B), is equal to the product of their individual probabilities: P(A) multiplied by P(B). In the context of the problem under discussion, each of the random variables X_i is independent and has the same uniform distribution, which allows us to multiply their probabilities to find the probability of their collective maximum being less than or equal to some value x.

Uniform Distribution

The uniform distribution is another cornerstone of probability theory. A random variable X is said to have a uniform distribution over an interval, for instance, (0, 1), if every number within that interval is equally likely to occur. The classic real-life example of a uniform distribution is a perfectly fair six-sided die, where each face, from 1 to 6, has an equal chance of landing face up.

In uniform distribution, the cumulative distribution function (CDF) is particularly easy to visualize: it increases linearly from 0 to 1 as x moves from the start to the end of the interval. Therefore, the CDF of a random variable X uniform on (0, 1) is simply x, as long as x falls within the interval. Outside of this interval, the probability is 0 for x less than 0 and 1 for x greater than 1. These properties are used to calculate the cumulative probability for the maximum of several such independent uniform variables in our exercise scenario.

Cumulative Distribution Function

The cumulative distribution function, often abbreviated as CDF, represents the probability that a random variable X takes on a value less than or equal to x. It's an incredibly powerful tool because it shows the entire range of probabilities in one function. For continuous random variables, the CDF is found by integrating the probability density function (PDF), and for discrete variables, it is the accumulation of probabilities at each point.

In the given exercise, we're analyzing the CDF of the maximum of several independent random variables with uniform distributions. The steps outlined in the solution demonstrate that the CDF of this maximum, denoted as F_M(x), is found by multiplying the CDFs of each individual random variable. Since we're dealing with independent uniform distributions, this results in raising x to the power of n, reflecting the number of variables. Understanding the relationship between the CDF and the PDF also allows us to derive the latter by differentiation, providing us with a complete picture of the behaviour of our maximum variable M within its distribution.