Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

An accountant wants to simplify his bookkeeping by rounding amounts to the nearest integer, for example, rounding \(€ 99.53\) and \(€ 100.46\) both to \(\in 100\). What is the cumulative effect of this if there are, say, 100 amounts? To study this we model the rounding errors by 100 independent \(U(-0.5,0.5)\) random variables \(X_{1}, X_{2}, \ldots, X_{100}\). a. Compute the expectation and the variance of the \(X_{i}\). b. Use Chebyshev's inequality to compute an upper bound for the probability \(\mathrm{P}\left(\left|X_{1}+X_{2}+\cdots+X_{100}\right|>10\right)\) that the cumulative rounding error \(X_{1}+\) \(X_{2}+\cdots+X_{100}\) exceeds \(€ 10\).

Step-by-Step Solution

Answer

Expectation: 0; Variance: \(\frac{1}{12}\); Upper bound: \(\frac{1}{12}\).

Step 1: Determine the expectation of each random variable

Each variable \(X_i\) follows a uniform distribution on the interval \([-0.5, 0.5]\). For a uniform distribution \(U(a, b)\), the expectation is given by \(E[X_i] = \frac{a+b}{2}\). Here \(a = -0.5\) and \(b = 0.5\), so we have \[ E[X_i] = \frac{-0.5 + 0.5}{2} = 0. \]

Step 2: Calculate the variance of each random variable

The variance of a uniform distribution \(U(a, b)\) is given by \(\text{Var}(X_i) = \frac{(b-a)^2}{12}\). Substituting \(a = -0.5\) and \(b = 0.5\), we get \[ \text{Var}(X_i) = \frac{(0.5 - (-0.5))^2}{12} = \frac{1^2}{12} = \frac{1}{12}. \]

Step 3: Calculate the total variance for 100 variables

Since the \(X_i\) are independent, the variance of the sum \(X_1 + X_2 + \cdots + X_{100}\) is the sum of their variances. So, \[ \text{Var}(X_1 + X_2 + \cdots + X_{100}) = \sum_{i=1}^{100} \text{Var}(X_i) = 100 \times \frac{1}{12} = \frac{100}{12} = \frac{25}{3}. \]

Step 4: Apply Chebyshev's inequality

Chebyshev's inequality states that for any random variable \(Y\) with mean \(\mu\) and variance \(\sigma^2\), and for any \(k > 0\), \[ P(|Y - \mu| \geq k) \leq \frac{\sigma^2}{k^2}. \] Here, \(Y = X_1 + X_2 + \cdots + X_{100}\), so \(\mu = 0\) and \(\sigma^2 = \frac{25}{3}\). We want \(P(|Y| > 10)\), so \(k = 10\). Thus, \[ P(|Y| > 10) \leq \frac{\frac{25}{3}}{10^2} = \frac{25}{3 \times 100} = \frac{25}{300} = \frac{1}{12}. \]

Key Concepts

Expected Value

Expected value is a fundamental concept in probability theory, representing the average outcome of a random variable if the experiment it models were repeated numerous times. It helps in predicting long-term results and is often denoted by the symbol \(E[X]\).

For a random variable with a uniform distribution, such as \(U(a, b)\), the expected value is calculated using the formula \(E[X] = \frac{a+b}{2}\).

In the current exercise, where each \(X_i\) follows \(U(-0.5, 0.5)\), the endpoints of our interval are \(-0.5\) and \(0.5\). By applying the formula, we find that:

• The expected value of each rounding error \(X_i\) is \(E[X_i] = \frac{-0.5 + 0.5}{2} = 0\).
• This implies, on average, rounding does not bias the total amount up or down over time.

Variance

Variance is a measure of how much the values of a random variable spread out from the expected value. A higher variance means more variability from the expected value. For a uniform distribution \(U(a, b)\), variance is determined using the formula \(\text{Var}(X) = \frac{(b-a)^2}{12}\).

Considering our context of independent uniform random variables \(X_i\) on the interval \([-0.5, 0.5]\), the endpoints suggest:

• Variance of single variable: \(\text{Var}(X_i) = \frac{(0.5 - (-0.5))^2}{12} = \frac{1}{12}\).
• Total variance for the sum of 100 such variables: sum variances since they are independent \(= 100 \times \frac{1}{12} = \frac{25}{3}\).

Variance reflects not bias but the potential deviation or spread around that expected value of zero for multiple trials.

Chebyshev's Inequality

Chebyshev's inequality is a key tool in probability theory that provides bounds on the probability that a random variable differs from its mean. With an assumption of no specific distribution (apart from having a finite variance), it states that for any random variable \(Y\) with mean \(\mu\) and variance \(\sigma^2\), and for any \(k > 0\),

\[ P(|Y - \mu| \geq k) \leq \frac{\sigma^2}{k^2}. \]

In our exercise, this inequality helps to bound the probability that the cumulative rounding error \(Y = X_1 + X_2 + \cdots + X_{100}\) exceeds \(10\).

• Given, \(\mu = 0\) and \(\sigma^2 = \frac{25}{3}\).
• We want \(P(|Y| > 10)\), thus \(k = 10\): \(P(|Y| > 10) \leq \frac{\frac{25}{3}}{10^2} = \frac{1}{12}\).

This calculation indicates the probability of the total rounding error being more than \(€10\) is quite low.

Uniform Distribution

A uniform distribution is one where all outcomes in a given range are equally likely. It can be either discrete or continuous, defined by the interval \([a, b]\). In our problem, the rounding errors \(X_i\) have a continuous uniform distribution within \([-0.5, 0.5]\).

Key characteristics of a continuous uniform distribution include:

• All intervals of the same length within \([a, b]\) have an equal probability.
• The mean or expected value is \(\frac{a+b}{2}\).
• The variance is \(\frac{(b-a)^2}{12}\), which impacts the spread of the distribution.

Uniform distributions are simple yet powerful models for many practical scenarios such as modeling errors that are equally likely to be positive or negative, like rounding errors.