Problem 28 - Calculus for Biology and Medicine

Question

Let \(X\) and \(Y\) be two random variables with the following joint distribution: $$ \begin{array}{ccc} \hline & X=0 & X=1 \\ \hline \boldsymbol{Y}=\mathbf{0} & 0.2 & 0.0 \\ \boldsymbol{Y}=\mathbf{1} & 0.3 & 0.5 \\ \hline \end{array} $$ (a) Find \(P(X=0, Y=1)\). (b) Find \(P(X=0)\). (c) Find \(P(Y=1)\). (d) Find \(P(X=0 \mid Y=0)\).

Step-by-Step Solution

Answer

(a) 0.3, (b) 0.5, (c) 0.8, (d) 1.

Step 1: Find P(X=0, Y=1)

The probability of the event where both conditions \(X=0\) and \(Y=1\) occur simultaneously is directly provided in the joint distribution table. Locate the cell corresponding to \(X=0\) and \(Y=1\). From the table, \(P(X=0, Y=1)=0.3\).

Step 2: Find P(X=0)

The probability \(P(X=0)\) is found by summing the probabilities of all events where \(X=0\), regardless of \(Y\). From the table, add \(P(X=0, Y=0) = 0.2\) and \(P(X=0, Y=1) = 0.3\). So, \(P(X=0) = 0.2 + 0.3 = 0.5\).

Step 3: Find P(Y=1)

To find \(P(Y=1)\), sum the probabilities of all events where \(Y=1\), regardless of \(X\). From the table, add \(P(X=0, Y=1) = 0.3\) and \(P(X=1, Y=1) = 0.5\). Thus, \(P(Y=1) = 0.3 + 0.5 = 0.8\).

Step 4: Find P(X=0 | Y=0)

The conditional probability \(P(X=0 | Y=0)\) is computed using the formula \(P(X=0 | Y=0) = \frac{P(X=0, Y=0)}{P(Y=0)}\). First, find \(P(Y=0)\) by summing probabilities where \(Y=0\): \(P(X=0, Y=0) = 0.2\) and \(P(X=1, Y=0) = 0.0\), so \(P(Y=0) = 0.2\). Then, \(P(X=0 | Y=0) = \frac{0.2}{0.2} = 1\).

Key Concepts

Conditional Probability: Understanding Dependence

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This is crucial when dealing with related events, where the occurrence of one event changes our expectations about another event. Mathematically, the conditional probability of event A given event B is given by the formula: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \] This formula essentially refines the sample space to just those outcomes where event B has occurred and then calculates the likelihood of event A happening within those confines. Some important aspects to remember about conditional probability:

• It helps in understanding how different events influence each other.
• If two events are independent, the conditional probability doesn't change anything. For example, tossing a fair coin doesn't influence the roll of a die.
• In our exercise, calculating \(P(X=0 \mid Y=0)\) is a typical example where this concept is crucial. Here, we're interested in knowing the probability of \(X=0\) given that \(Y=0\) has already happened.

Random Variables: Assigning Values to Outcomes

A random variable is a concept used to map outcomes of a random process to numbers. Essentially, it is a function that assigns numeric values to each outcome in a sample space. There are two types of random variables: discrete and continuous. In our context, we'll focus on discrete random variables, which take on a countable number of distinct possibilities. Key aspects of discrete random variables:

• They can assume values like 0, 1, 2, and so on, depending on the scenario.
• The probability of each value is defined by a probability distribution.
• In our exercise, \(X\) and \(Y\) are discrete random variables representing specific outcomes in the probability distribution table.
• Each cell in the probability distribution table shows the probability of a particular combination of \(X\) and \(Y\) values occurring.

Thus, random variables are the bridges linking abstract probabilistic assumptions to number-based representations, helping us make sense of possible outcomes.

Probability Distribution Table: Visualizing Data Relationships

A probability distribution table provides a convenient way to visualize the joint probabilities of different outcomes of random variables. This table helps us examine how random variables relate to each other. Each cell in the table represents the joint probability of specific values of the random variables. In our exercise, the table helps track probabilities of \(X\) and \(Y\) as they occur together:

• The rows and columns correspond to the values taken by the random variables, in this case, \(X\) and \(Y\).
• The intersection of each row and column shows the joint probability of those values.
• Adding values across a row or down a column allows us to find marginal probabilities, like \(P(X=0)\) or \(P(Y=1)\).
• This layout makes it easier to perform calculations, such as finding correlated outcomes or conditional probabilities.

Thus, probability distribution tables are essential tools for organizing and analyzing the relationship between random variables, making complex interactions transparent and easier to comprehend.