Problem 27 - Calculus for Biology and Medicine

Question

Let \(X\) and \(Y\) be two random variables with the following joint distribution: $$ \begin{array}{ccc} \hline & \boldsymbol{X}=\mathbf{0} & \boldsymbol{X}=\mathbf{1} \\ \hline \boldsymbol{Y}=\mathbf{0} & 0.3 & 0.1 \\ \boldsymbol{Y}=\mathbf{1} & 0.2 & 0.4 \\ \hline \end{array} $$ (a) Find \(P(X=1, Y=0)\). (b) Find \(P(X=1)\). (c) Find \(P(Y=0)\). (d) Find \(P(Y=0 \mid X=1)\).

Step-by-Step Solution

Answer

(a) 0.1; (b) 0.5; (c) 0.4; (d) 0.2.

Step 1: Identify the Joint Probability

The joint probability \( P(X=1, Y=0) \) refers to the probability that \( X = 1 \) and \( Y = 0 \) occur simultaneously. From the joint distribution table, we find this value at the intersection of \( X = 1 \) and \( Y = 0 \), which is 0.1.

Step 2: Determine the Probability of X=1

To find \( P(X=1) \), sum all probabilities where \( X = 1 \). From the table, these are \( P(X=1, Y=0) = 0.1 \) and \( P(X=1, Y=1) = 0.4 \). So, \( P(X=1) = 0.1 + 0.4 = 0.5 \).

Step 3: Compute the Probability of Y=0

To find \( P(Y=0) \), sum all probabilities where \( Y = 0 \). From the table, these are \( P(X=0, Y=0) = 0.3 \) and \( P(X=1, Y=0) = 0.1 \). So, \( P(Y=0) = 0.3 + 0.1 = 0.4 \).

Step 4: Calculate the Conditional Probability P(Y=0 | X=1)

The conditional probability \( P(Y=0 | X=1) \) is given by \( \frac{P(X=1, Y=0)}{P(X=1)} \). From earlier steps, \( P(X=1, Y=0) = 0.1 \) and \( P(X=1) = 0.5 \). Thus, \( P(Y=0 | X=1) = \frac{0.1}{0.5} = 0.2 \).

Key Concepts

Conditional Probability

Conditional probability helps us understand the likelihood of an event occurring if another related event is known to have already taken place. It is represented as \( P(A \mid B) \), which reads "the probability of \( A \) given \( B \)." This concept allows us to focus on a subset of possible outcomes that affect the event's probability.
For example, from the original exercise, we calculated the conditional probability \( P(Y=0 \mid X=1) \). Here, \( X=1 \) is the given condition, which refines our focus to only the scenarios where \( X = 1 \). We found that this probability is \( \frac{0.1}{0.5} = 0.2 \). This means, given that \( X \) equals 1, the chance of \( Y \) being 0 is 20%.

• The formula for calculating conditional probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] where \( P(A \cap B) \) is the joint probability that both \( A \) and \( B \) occur, and \( P(B) \) is the probability of the given event.
• Conditional probability is important in statistical inference, allowing predictions and insights in various fields such as finance, medicine, and weather forecasting.

Marginal Probability

Marginal probability refers to the probability of a single event occurring without considering the influence or outcome of another variable. Think of it as the "total" probability of just one variable.
In the exercise, we explored how to find marginal probabilities like \( P(X=1) \) and \( P(Y=0) \). We sum up probabilities across the possible outcomes of the other variable. For example:

• To find \( P(X=1) \), the probabilities \( P(X=1, Y=0) \) and \( P(X=1, Y=1) \) are added together, giving us \( 0.5 \).
• Similarly, to find \( P(Y=0) \), we sum \( P(X=0, Y=0) + P(X=1, Y=0) = 0.4 \).

This process shows how marginal probabilities are computed by considering all scenarios for the variable of interest, independent of any other variable’s specific outcome. Marginal probabilities help us get an overall understanding of each variable's behavior in a dataset.

Random Variables

Random variables are foundational in probability theory, representing quantities with uncertain outcomes. They are often denoted by capital letters like \( X \) and \( Y \), each having its probability distribution.

• A random variable can be classified as discrete or continuous. Discrete random variables, like in our exercise, take on a finite set of values. Continuous ones can assume any value within a range.
• Every random variable is associated with a probability distribution, which gives the probabilities of its various potential outcomes. For the given table, we have two random variables, \( X \) and \( Y \), with values 0 and 1. Their joint distribution displays all possible combinations of these values along with their associated probabilities.
• Understanding random variables and their distributions helps us model real-world phenomena, conduct hypothesis testing, and make predictions.

Random variables, with their distributions, provide a framework for statistical analysis, highlighting the variability and uncertainty inherent in random processes.