Problem 37 - Applied Statistics and Probability for Engineers

Question

Patients are given a drug treatment and then evaluated. Symptoms either improve, degrade, or remain the same with probabilities \(0.4,0.1,0.5,\) respectively. Assume that four independent patients are treated and let \(X\) and \(Y\) denote the number of patients who improve or degrade. Are \(X\) and \(Y\) independent? Calculate the covariance and correlation between \(X\) and \(Y\).

Step-by-Step Solution

Answer

Yes, \( X \) and \( Y \) are independent with zero covariance and correlation.

Step 1: Define Random Variables

Let \( X \) be the random variable representing the number of patients who improve, and \( Y \) be the random variable representing the number of patients who degrade. Each patient has a probability of \( 0.4 \) of improving and \( 0.1 \) of degrading.

Step 2: Calculate Expected Values

The expected value of a binomial distribution is given by \( E[X] = np \). Here, for \( X \), \( n = 4 \) and \( p = 0.4 \), so \( E[X] = 4 \times 0.4 = 1.6 \). For \( Y \), \( p = 0.1 \), so \( E[Y] = 4 \times 0.1 = 0.4 \).

Step 3: Calculate Variances

The variance of a binomial distribution is \( \text{Var}(X) = np(1-p) \). For \( X \), \( \text{Var}(X) = 4 \times 0.4 \times 0.6 = 0.96 \). For \( Y \), \( \text{Var}(Y) = 4 \times 0.1 \times 0.9 = 0.36 \).

Step 4: Calculate Covariance

Covariance \( \text{Cov}(X, Y) \) involves understanding the joint PMF, but since each patient has independent outcomes, market-improving or degrading jointly doesn't affect others. The probability that a patient improves then degrades should be zero beyond the independence effect, thus, \( \text{Cov}(X, Y) = 0 \).

Step 5: Calculate Correlation

Correlation is given by \( \rho = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \). Since \( \text{Cov}(X, Y) = 0 \), \( \rho = \frac{0}{\sqrt{0.96 \cdot 0.36}} = 0 \).

Step 6: Determine Independence

Two variables are independent if their covariance is zero. Since \( \text{Cov}(X, Y) = 0 \), \( X \) and \( Y \) are independent.

Key Concepts

Understanding Probability

Probability is a measure of the likelihood that a particular event will occur. It is a fundamental concept in statistics used to quantify uncertainty. In this context, probability ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty. For the exercise given, each patient could either improve, degrade, or remain the same after a drug treatment.

• The probability that symptoms improve is 0.4.
• The probability that symptoms degrade is 0.1.
• The probability of symptoms remaining the same is 0.5.

These probabilities are independent events since each patient's outcomes do not affect other patients' results. All probabilities add up to 1, covering all possible outcomes. Understanding these probabilities helps in predicting and calculating statistical measures like expected outcomes.

Exploring Random Variables

A random variable is a numerical description of the outcome of a statistical experiment. In the given exercise, we define two random variables: \(X\) and \(Y\). Here, \(X\) represents the number of patients who experience an improvement in symptoms, and \(Y\) represents the number of patients who experience a degradation in symptoms.
Random variables can take on different values based on the outcomes of a chance event. For example, since we are dealing with four patients, \(X\) can potentially be any integer from 0 to 4, representing 0 to 4 patients showing improvement. Similarly, \(Y\) can also take values from 0 to 4, signifying the number of patients showing degraded symptoms.
Understanding how to define and work with random variables is crucial for performing statistical calculations such as expected values and variances.

Calculating Covariance

Covariance is a statistical measure that describes the extent to which two random variables change together. A positive covariance indicates that as one variable increases, the other tends to increase as well. Conversely, a negative covariance suggests that when one variable increases, the other decreases.
In our scenario, we calculated the covariance between \(X\) and \(Y\), which represent the number of patients whose symptoms improve and degrade, respectively. To determine this, the step-by-step solution explains the use of expected values and joint probabilities. However, since each patient's outcome is independent, the joint probability of simultaneous improvement and degradation is 0, resulting in a covariance of 0. This zero covariance implies no systematic relationship between the two variables under this treatment scenario.

Determining Independence in Statistics

Independence in statistical terms means that two events do not influence each other's outcomes. For random variables, it implies that the occurrence of one variable does not affect the probability distribution of another. In statistics, if the covariance between two random variables is zero, they are considered independent.
In the exercise provided, the covariance between \(X\) (number of patients improved) and \(Y\) (number of patients degraded) was found to be zero. This result confirms that the outcomes of improvement or degradation are independent events, meaning one does not affect the likelihood of the other.
Understanding independence is key in simplifying complex probability problems, as independent variables allow for easier computation due to separated probabilities.