Problem 27 - Calculus for Biology and Medicine

Question

Let \(X\) and \(Y\) be two random variables with the following joint distribution: $$\begin{array}{ccc} \hline & X=0 & X=1 \\ \hline Y=0 & 0.3 & 0.1 \\ Y=1 & 0.2 & 0.4 \\ \hline \end{array}$$ (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 P(X=1, Y=0)

Use the table to find the probability where \(X=1\) and \(Y=0\). From the table, the probability \(P(X=1, Y=0)\) is given directly as 0.1.

Step 2: Find P(X=1)

To find \(P(X=1)\), sum all the probabilities where \(X=1\). According to the table: \(P(X=1) = P(X=1, Y=0) + P(X=1, Y=1) = 0.1 + 0.4 = 0.5\).

Step 3: Find P(Y=0)

To find \(P(Y=0)\), sum all the probabilities where \(Y=0\). According to the table: \(P(Y=0) = P(X=0, Y=0) + P(X=1, Y=0) = 0.3 + 0.1 = 0.4\).

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

Use the formula for conditional probability: \(P(Y=0 \mid X=1) = \frac{P(X=1, Y=0)}{P(X=1)}\). Substituting the values found previously: \(P(Y=0 \mid X=1) = \frac{0.1}{0.5} = 0.2\).

Key Concepts

Joint Distribution

Joint distribution is a fundamental concept in probability and statistics. It refers to the probability distribution of two or more random variables occurring at the same time. In this exercise, the joint distribution is provided for random variables \(X\) and \(Y\). The table given shows probabilities of all combinations of \(X\) and \(Y\) values. For example, the entry for \((X=1, Y=0)\) is 0.1, meaning there is a 10% chance that \(X\) equals 1 and \(Y\) equals 0 simultaneously.

The joint distribution allows us to understand how the probabilities are distributed over different possible outcomes of the two random variables. By examining these combined probabilities, we can infer important statistical properties and relationships between \(X\) and \(Y\). With joint distributions, you can:

• Identify dependent relationships between random variables.
• Calculate the probability of events that involve more than one outcome.
• Derive marginal distributions by summing probabilities over the other variable.

Conditional Probability

Conditional probability gives us the probability of an event occurring given that another event has already occurred. It's an essential concept when dealing with joint distributions, as it provides insights into the relation between random variables.

In the context of this exercise, we find \(P(Y=0 | X=1)\), the probability that \(Y=0\) given \(X=1\). This is calculated using the formula:\[P(Y=0 \mid X=1) = \frac{P(X=1, Y=0)}{P(X=1)}\]We have already determined that \(P(X=1, Y=0) = 0.1\) and \(P(X=1) = 0.5\), thus:\[P(Y=0 \mid X=1) = \frac{0.1}{0.5} = 0.2\]

• This tells us that when \(X=1\), there is a 20% chance that \(Y\) is 0.
• Such calculations help in analyzing how likely a specific outcome is under a given condition.

Conditional probability is powerful in real-world scenarios as it helps in decision-making processes, predicting future events, and understanding causal relationships.

Random Variables

Random variables are variables whose possible values are numerical outcomes of a random phenomenon. In this exercise, \(X\) and \(Y\) are random variables representing two different outcomes. These variables follow a probability distribution which determines how likely each outcome is.

For instance, \(X\) can take values 0 or 1, and similarly, \(Y\) can be 0 or 1, as shown in the joint distribution table. Each combination of these values, such as \(X=1, Y=0\), has an associated probability.

• Random variables can be discrete, like in this example where \(X\) and \(Y\) have specific outcomes (0 or 1) with set probabilities.
• They provide a way to quantify uncertain events and can be used to create statistical models.
• Understanding random variables allows for deeper analysis into how likely certain scenarios are, based on their probability distributions.

Comprehending random variables involves knowing their nature (discrete or continuous) and how their probability distributions work, which in turn influences joint and conditional probabilities.