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Chapter 7: Problem 16

Let X and Y be the random variables that count the number of heads and the number of tails that come up when two fair coins are flipped. Show that X and Y are not independent.

Short Answer

Expert verified
X and Y are not independent because P(X = 1, Y = 1) = 1/2 does not equal P(X = 1)P(Y = 1) = 1/4.

Step by step solution

01

Define Random Variables

Let X be the random variable representing the number of heads and Y be the random variable representing the number of tails when two fair coins are flipped. Therefore, X and Y can take values from the set {0, 1, 2}.
02

Determine Joint Distribution

List all possible outcomes for flipping two coins: (HH), (HT), (TH), (TT). Calculate the (X, Y) pairs for each outcome: (2, 0) for (HH), (1, 1) for (HT) and (TH), and (0, 2) for (TT).
03

Calculate Marginal Distributions

Find the probability for each value of X and Y separately. For X: P(X = 0) = 1/4, P(X = 1) = 1/2, P(X = 2) = 1/4. For Y: P(Y = 0) = 1/4, P(Y = 1) = 1/2, P(Y = 2) = 1/4.
04

Calculate Joint Probabilities

Determine the joint probability P(X = x, Y = y) by counting occurrences in all outcomes. For example, P(X = 1, Y = 1) = 2/4 = 1/2 based on events (HT) and (TH).
05

Check Independence Condition

Check if P(X = x, Y = y) = P(X = x)P(Y = y) holds for all pairs. For example, P(X = 1, Y = 1) = 1/2 but P(X = 1)P(Y = 1) = 1/2 * 1/2 = 1/4. Since these are not equal, X and Y are not independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Joint Distribution
When studying random variables like X and Y, one useful concept is the joint distribution. The joint distribution gives the probability of different pairs (X, Y), allowing us to see how these variables move together. For instance, when flipping two fair coins, there are four possible outcomes: (HH), (HT), (TH), and (TT). The counts of heads and tails, represented by the pairs (X, Y), help us create a joint distribution. For our example, this would look like: (2, 0) for (HH), (1, 1) for (HT) and (TH), and (0, 2) for (TT). By calculating the probabilities of these pairs, students can start understanding the idea of how two random variables relate.
Marginal Distribution
Once we've built our joint distribution, we can also look at the marginal distribution. This will help us understand each random variable on its own, without considering the other. To do this, we simply sum the joint probabilities for all ranges of the other variable. For our coin example, the marginal distribution of X (number of heads) can be found by adding relevant joint probabilities. This gives us: P(X = 0) = 1/4, P(X = 1) = 1/2, and P(X = 2) = 1/4. The same process applies for Y (number of tails), giving the same probabilities. Understanding marginal distributions is fundamental for grasping more complex probabilistic concepts.
Independence of Random Variables
Two random variables are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, X and Y are independent if for all x and y, P(X = x, Y = y) = P(X = x)P(Y = y). This means that the joint probability is equal to the product of their marginal probabilities. In our example with the coins, we assess whether P(X = 1, Y = 1) is equal to P(X = 1)P(Y = 1). We find that P(X = 1, Y = 1) = 1/2 but P(X = 1)P(Y = 1) = (1/2)(1/2) = 1/4. Since these don't match, we conclude that X and Y are not independent.
Probability Calculations
Probability calculations involve various methods and rules to determine the likelihood of events. To fully understand our coin flip example, we perform several types of calculations:
  • Calculating joint probabilities: By counting occurrences of (X, Y) pairs.
  • Calculating marginal probabilities: By summing joint probabilities for specific X or Y values.
  • Checking for independence: By comparing P(X = x, Y = y) to P(X = x)P(Y = y).
These calculations deepen our understanding of how random variables behave and help predict outcomes. Practice makes perfect鈥攕o keep working on similar problems to enhance your skills!

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