Problem 61 - An Introduction to Mathematical Statistics and Its Applications

Question

Let the random variable \(X\) take on the values \(1,2, \ldots, n\), each with probability \(1 / n .\) Define \(Y\) to be \(X^{2} .\) Find \(\rho(X, Y)\) and \(\lim _{n \rightarrow \infty} \rho(X, Y)\).

Step-by-Step Solution

Answer

The correlation coefficient \(\rho(X, Y)\) can be found by substituting the calculated values into its formula and simplifying. The limit of \(\rho(X, Y)\) as \(n\) approaches infinity can be found by plugging in \(\infty\) for \(n\) in the \(\rho(X, Y)\) formula, using l'Hospital's rule if necessary and simplifying.

Step 1: Definition of correlation

The correlation coefficient \(\rho(X, Y)\) is defined by \(\rho(X,Y) = \frac{E[XY] - E[X]E[Y]}{\sqrt{Var[X]Var[Y]}}\). So our first aim is to calculate each of these values.

Step 2: Calculate E[X] and E[Y]

First, given that the probability of each value of \(X\) is \(1/n\), the expected value of \(X\) is \(E[X] = \frac{1}{n}\sum_{i=1}^{n}i = \frac{n(n+1)}{2n} = \frac{n+1}{2}\). Similarly, for \(Y\), since \(Y = X^{2}\), we have \(E[Y] = \frac{1}{n}\sum_{i=1}^{n}i^{2} = \frac{n(n+1)(2n+1)}{6n} = \frac{(n+1)(2n+1)}{6}\).

Step 3: Calculate E[XY]

Next, \(E[XY]=E[X*X^{2}]=E[X^{3}]=\frac{1}{n}\sum_{i=1}^{n}i^{3} = \frac{n^{2}(n+1)^{2}}{4n} =\frac{n(n+1)^{2}}{4}\).

Step 4: Calculate Var[X] and Var[Y]

The variance \(Var[X] = E[X^{2}] -(E[X])^{2}= \frac{(n+1)(2n+1)}{6} - \left(\frac{n+1}{2}\right)^{2}\). Similarly, for \(Y\), \(Var[Y] = E[Y^{2}] - (E[Y])^{2}\). But here, \(Y^{2} = X^{4}\), and \(E[Y^{2}] = E[X^{4}] = \frac{1}{n}\sum_{i=1}^{n}i^{4} =\frac{n(n+1)(2n+1)(3n^{2}+3n-1)}{30n}= \frac{(n+1)(2n+1)(3n^{2}+3n-1)}{30}\), so \(Var[Y] = \frac{(n+1)(2n+1)(3n^{2}+3n-1)}{30} - \left(\frac{(n+1)(2n+1)}{6}\right)^{2}\).

Step 5: Calculate ρ(X, Y)

Substitute the values calculated in steps 2, 3, and 4 to find the correlation coefficient \(\rho(X, Y)\).

Step 6: Find the limit as n approaches infinity

To find the limit as \(n\) approaches infinity of \(\rho(X, Y)\), plug in \(\infty\) for \(n\) in the expression \(\rho(X, Y)\). Use L'Hospital's rule if necessary and simplify.

Key Concepts

Expected Value

The concept of the expected value is fundamental in the field of probability and statistics; it is a measure of the central tendency of a random variable. If we think of it in terms of a game or experiment, the expected value is essentially the average outcome we would anticipate after many repetitions of the game or experiment.

For a discrete random variable, which can take on a number of distinct values (like rolling a die with six sides), the expected value can be calculated by multiplying each potential value by its probability and then adding all these products together. Mathematically, this is represented as: \
E[X] = \sum_{i=1}^{n} p_{i} * x_{i}\
, where \(x_{i}\) is a possible value of \(X\), and \(p_{i}\) is the probability of \(X\) taking on the value \(x_{i}\). In our exercise, \(X\) takes on the values \(1, 2, ..., n\), each with probability \(1/n\), leading to an expected value calculation of \(E[X] = \frac{n+1}{2}\).

Variance

Moving on from the average of a distribution to its spread, we encounter the concept of variance. Variance quantifies how much a set of numbers (in our case, the possible values of a random variable) are spread out from their expected value. A low variance indicates that the numbers are close to the expected value, while a high variance means they are more spread out.

Formally, the variance of a random variable \(X\) is given by \
Var[X] = E[X^{2}] - (E[X])^{2}\
, which is the expected value of the squared deviations from the mean. For the random variable in our exercise, the variance for \(X\) is calculated by first finding the expected value of \(X^{2}\), then subtracting the square of \(E[X]\), as shown in Step 4 of the solution. By providing an understanding of the variability of a random variable, variance is a key measure in assessing risk and variability in data. This makes variance particularly relevant in fields such as finance and engineering, where making predictions with certainty is critical.

Limit of a Function

When exploring functions in calculus, the limit of a function is encountered as a way of describing the behavior of functions as they approach a certain point. It tells us the value that a function approaches as the input (or the independent variable) approaches some value.

Limits are foundational to the study of calculus and are used to define continuous functions, derivatives, and integrals. For example, the limit \
\lim_{x \rightarrow c} f(x)\
expresses the value that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(c\). If the function approaches the same value from both the left and right as \(x\) approaches \(c\), then the limit exists at that point.

In the context of our exercise, calculating the limit as \(n\) approaches infinity of the correlation coefficient (\(\rho(X, Y)\)) allows us to understand the long-term behavior of the relationship between \(X\) and \(Y\). This provides insight into whether the variables become more or less correlated as \(n\) grows large. Finding this limit may involve advanced techniques such as L'Hospital's Rule, especially when dealing with indeterminate forms like \(0/0\) or \(\infty/\infty\).