Problem 8 - Calculus for Biology and Medicine

Question

Suppose \(X\) is a random variable with mean \(-5\) and variance 2\. What can you say about the probability that \(X\) deviates from its mean by at least 4 ?

Step-by-Step Solution

Answer

The probability that \( X \) deviates from its mean by at least 4 is at most 0.125.

Step 1: Identify the given information

We are given a random variable \( X \) with a mean \( \mu = -5 \) and a variance \( \sigma^2 = 2 \). The problem asks about the probability that \( X \) deviates from its mean by at least 4.

Step 2: Introduce Chebyshev's Inequality

Chebyshev's Inequality is a statistical theorem that provides a bound on the probability that the value of a random variable deviates from its mean. It is stated as: \[ P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2} \] where \( \sigma \) is the standard deviation of \( X \), and \( k \) is the number of standard deviations away from the mean.

Step 3: Calculate the standard deviation

The standard deviation \( \sigma \) is the square root of the variance. Given \( \sigma^2 = 2 \), we calculate \( \sigma = \sqrt{2} \approx 1.414 \).

Step 4: Set up for Chebyshev's Inequality

We need to determine \( k \) such that \( k\sigma = 4 \). Thus, \( k = \frac{4}{\sigma} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \approx 2.828 \).

Step 5: Apply Chebyshev's Inequality

Substitute \( k \) into the inequality: \[ P(|X - \mu| \geq 4) \leq \frac{1}{(2\sqrt{2})^2} = \frac{1}{8} = 0.125 \] Thus, the probability that \( X \) deviates from its mean by at least 4 is at most 0.125.

Key Concepts

Random Variable

A random variable is a fundamental concept in probability and statistics, representing a quantity whose value is subject to variations due to chance. It is essentially a function that assigns a numerical value to each event in a sample space.

Random variables are crucial in the realm of probability as they allow us to quantify and analyze random phenomena. They are categorized into two main types:

• Discrete Random Variables: These are variables that can take on a countable number of distinct values. Examples include the roll of a dice (1, 2, 3, 4, 5, or 6) or the flip of a coin (heads or tails).
• Continuous Random Variables: These variables can take on an infinite number of possible values within a given range, such as the temperature on a specific day or the time it takes to run a race.

The random variable in our example, denoted as \(X\), is associated with a mean \(\mu = -5\) and variance, which allows us to apply concepts like Chebyshev's Inequality to predict probabilities related to its values.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values. It is the square root of the variance and provides insight into the spread of the data around the mean.

When discussing a random variable, the standard deviation helps us understand how much the values deviate on average from the mean value \(\mu\). It is symbolized by \(\sigma\) in formulas.

• The smaller the standard deviation, the closer the data points are to the mean.
• A larger standard deviation indicates that the data points are spread out over a wider range.

For our random variable \(X\) with variance \(\sigma^2 = 2\), the standard deviation is calculated as \(\sigma = \sqrt{2} \approx 1.414\). This value is used in Chebyshev's Inequality to determine the likelihood that \(X\) deviates significantly from its mean.

Variance

Variance is another critical concept in statistics that measures the spread of a set of numbers. It quantifies how far each number in the set is from the mean and thus from every other number in the set.

The variance is calculated by taking the average of the squared differences from the mean. In mathematical terms, if \(X\) has a mean \(\mu\), then the variance \(\sigma^2\) is given by:\[\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2\]where \(N\) is the number of data points.

• A small variance indicates that the data points tend to be very close to the mean and hence to each other.
• A larger variance signifies that the data points are spread out over a larger range of values.

In our scenario, the variance of the random variable \(X\) is 2, which is a key piece of information for using Chebyshev's Inequality to understand the variability and distribution behavior of \(X\).