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 is at most \(\frac{1}{8}\).

Step 1: Identify the Given Parameters

We are given that the mean of the random variable \(X\) is \(-5\) and the variance is \(2\). The standard deviation \(\sigma\) is the square root of the variance, so it is \(\sqrt{2}\).

Step 2: Express the Deviation Condition

We need to find the probability that the absolute deviation \(|X + 5|\) is greater than or equal to 4. This is equivalent to calculating \(P(|X - (-5)| \geq 4)\).

Step 3: Apply Chebyshev's Inequality

Chebyshev's inequality states that for any random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\), \(P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}\) for \(k > 0\).

Step 4: Calculate the Bound Using Chebyshev's Inequality

In this context, \(k\) can be calculated as \(\frac{4}{\sqrt{2}} = 2\sqrt{2}\). According to Chebyshev's inequality, \(P(|X - (-5)| \geq 4) \leq \frac{1}{(2\sqrt{2})^2} = \frac{1}{8}\).

Step 5: Interpret the Result

The probability that \(X\) will deviate from its mean by at least 4 is at most \(\frac{1}{8}\), or 0.125.

Key Concepts

Understanding Random Variables

A random variable is a foundational concept in probability and statistics that helps us quantify outcomes of a random phenomenon. Imagine rolling a dice. The result—whether it's a 1, 2, or up to 6—is what we call a random variable. This result can vary every time you roll. Random variables are crucial because they connect real-life random situations to mathematical expressions.
There are two types of random variables you might encounter:

• Discrete Random Variables: These have a countable number of possible values. For example, the number of heads when flipping a coin multiple times.
• Continuous Random Variables: These can take any value within a given range. For instance, the exact time it takes to complete a 100-meter race.

Understanding random variables is key to navigating more complex topics in statistics and probability, setting the stage for concepts like mean and variance.

Exploring Mean and Variance

The mean and variance are critical statistical elements that offer insights into a random variable's behavior.

• Mean: Often called the average, the mean is a measure of the central tendency of a random variable. It gives a sense of where most of the data points fall. In the provided exercise, the mean of the random variable X is -5. This implies that, on average, the outcomes tend to cluster around this value.
• Variance: While the mean gives a central location, the variance tells us how spread out the values are from the mean. A larger variance indicates that the data points are more spread out. In our case, the variance is 2, indicating moderate spread around the mean of -5.

These concepts work hand in hand to depict the distribution of outcomes of a random variable, forming the basis for further probability calculations, such as in Chebyshev's Inequality which leverages these attributes for broader understanding.

Grasping Standard Deviation

The standard deviation is a statistical measure that builds on the variance to provide a clear, interpretable measure of spread or dispersion of a dataset or random variable. Simply put, while variance is informative, standard deviation gives this information in the same units as the data, making it easier to understand.

• Calculation: The standard deviation is the square root of the variance. In our example, the variance is 2, so the standard deviation is \( \sqrt{2} \), simplifying to about 1.41.
• Interpretation: This value tells us that, on average, the values of the random variable X deviate from the mean (-5) by approximately 1.41 units. Standard deviation thus quantifies variability and helps in comparing different datasets.

In practice, standard deviation is pivotal. It allows statisticians and data analysts to assess data patterns, make predictions, and apply inequalities like Chebyshev's Inequality to determine the likelihood of deviations beyond typical variations.