Problem 22 - A Computational Introduction to Number Theory and Algebra

Question

Let \(X\) be a \(0 / 1\) -valued random variable. Show that \(\operatorname{Var}[X] \leq 1 / 4\).

Step-by-Step Solution

Answer

Question: Prove that the variance of a 0/1-valued random variable X is bounded by 1/4. Answer: To prove this statement, first, we computed the expected value of X and X^2, which were found to be equal to p in both cases, where p is the probability of X=1. Next, we calculated the variance of X as Var[X] = p - p^2. Then, we showed that p - p^2 ≤ 1/4 by representing it as a non-negative quadratic inequality: (p - 1/2)^2 ≥ 0. Since the inequality holds true, we can conclude that the variance of the 0/1-valued random variable X is bounded by 1/4.

Step 1: Compute the expected value of X

To compute the expected value of \(X\), which is denoted as \(E[X]\), we need to consider the probabilities of \(X\) taking the values 0 and 1. Let's denote the probability of \(X=1\) as \(p\), so the probability of \(X=0\) is \(1-p\). Since \(X\) can only take the values 0 and 1, we can compute the expected value as follows: $$E[X] = 0 \times (1-p) + 1 \times p = p.$$

Step 2: Compute the expected value of X^2

Since \(X\) can only take the values of 0 and 1, we know that \(X^2 = X\). Therefore, the expected value of \(X^2\) is the same as the expected value of \(X\): $$E[X^2] = E[X] = p.$$

Step 3: Compute the variance of X

We can now compute the variance of \(X\), denoted as \(\operatorname{Var}[X]\) using the following formula: $$\operatorname{Var}[X] = E[X^2] - E[X]^2.$$ Substituting our calculated values of \(E[X]\) and \(E[X^2]\): $$\operatorname{Var}[X] = p - p^2.$$

Step 4: Show that Var[X] <= 1/4

We want to show that \(\operatorname{Var}[X] = p - p^2 \leq \frac{1}{4}\). To do this, we focus on the inequality: $$p - p^2 \leq \frac{1}{4}.$$ To solve the inequality, first, rewrite it as a quadratic inequality: $$p^2 - p + \frac{1}{4} \geq 0.$$ Notice that the inequality factors as: $$(p - \frac{1}{2})^2 \geq 0.$$ Since the square of any real number is non-negative, the inequality holds true. Therefore, the variance of the 0/1-valued random variable \(X\) is bounded by \(\frac{1}{4}\), as desired.

Key Concepts

0/1 Random Variable

A 0/1 random variable is a special type of random variable that can only take on two values: 0 and 1. This simplification often helps in statistical modeling and probability theory. These variables are fundamental in many areas, such as computer science and statistics, because they represent binary events, like success/failure or yes/no outcomes.Here are some key points about 0/1 random variables:

• They model dichotomous events.
• The probability of obtaining 1 is denoted as \(p\), while the probability of obtaining 0 is \(1-p\).
• The use of only two possible outcomes simplifies calculations and makes these variables particularly valuable in Bernoulli trials.

Due to these properties, 0/1 random variables are frequently used in algorithms and statistical models, laying foundational concepts for understanding discrete variables and their variance.

Expected Value

The expected value of a random variable, often referred to as its mean, is a critical concept that provides insight into the average outcome of a random process.For a 0/1 random variable \(X\), which takes the value 1 with probability \(p\) and 0 with probability \(1-p\), the expected value \(E[X]\) is calculated as:\[ E[X] = 0 \times (1-p) + 1 \times p = p. \]This formula showcases that for 0/1 variables, the expected value is simply the probability of the outcome being 1.Some important aspects to consider about expected value include:

• It's the weighted average of all possible values the variable can take, weighted by their probabilities.
• In terms of prediction, it gives a central tendency or the "average" in a probabilistic sense.
• It serves as a basis for calculating other characteristics, such as variance.

The concept of expected value is a cornerstone in probability and is particularly helpful for analyzing and predicting outcomes in situations modeled by random variables.

Quadratic Inequality

A quadratic inequality involves a squared term and can often be utilized to limit or bound certain values. In the context of variance calculation, we use quadratic inequalities to demonstrate that the variance of a 0/1 random variable \(X\), with variance \( \operatorname{Var}[X] = p - p^2 \), does not exceed \( \frac{1}{4} \).The key steps in handling such inequalities are:

• Convert the variance expression into a quadratic inequality: \( p^2 - p + \frac{1}{4} \geq 0 \).
• Recognize factorable forms: Factoring gives \( (p - \frac{1}{2})^2 \geq 0 \).
• Interpret the result: Since any square is non-negative, the inequality naturally holds.

Quadratic inequalities like these play a crucial role in establishing bounds for probabilities, means, and variances, demonstrating the non-negativity of certain expressions, and ensuring the validity of statistical assumptions.