Master the Combination of Random Variables in IB Math AI HL

Table of Contents

1. Introduction to the concept of combining random variables
2. Formulas for finding expected value and variance of combined random variables
3. Example 1: Combining random variables following a normal distribution
4. Example 2: Combining random variables following a Poisson distribution
5. Step-by-step solution for Example 1
6. Step-by-step solution for Example 2
7. Practice questions in the RV Question Bank
8. Conclusion and further practice recommendations

Introduction to the concept of combining random variables

When working with statistics and probability, one important concept to understand is the combination of random variables. This concept involves adding and subtracting two or more independent random variables to Create a new random variable that represents their combination. In this article, we will explore the formulas, examples, and step-by-step solutions for combining random variables following both a normal distribution and a Poisson distribution.

Formulas for finding expected value and variance of combined random variables

To find the expected value (mean) and variance of a combination of n independent random variables, we can use the following formulas:

1. Expected value formula:

   • E(X₁ + X₂ + ... + Xn) = E(X₁) + E(X₂) + ... + E(Xn)

2. Variance formula:

   • Var(X₁ + X₂ + ... + Xn) = Var(X₁) + Var(X₂) + ... + Var(Xn)

Additionally, when combining two random variables that follow a Poisson distribution, we can simply add their lambda values together to find the combined lambda.

Example 1: Combining random variables following a normal distribution

Consider a car wash where the time taken to clean a car follows a normal distribution with a mean of 50 minutes and a standard deviation of 4.5 minutes. Suppose two cars come in one after the other, and we want to determine the probability that the total time taken to clean both cars is 1.5 hours or less.

To solve this problem, we need to create a new random variable that represents the total time for two cars. We can do this by combining the mean and standard deviation of the individual random variables. Using our knowledge of the normal distribution, we can then calculate the probability.

Example 2: Combining random variables following a Poisson distribution

Take the Scenario of two cafes, Paris Cafe and Sens Cafe, where the number of customers entering each cafe follows a Poisson distribution. The lambda value for Paris Cafe is 210 and for Sens Cafe is 295. We Are asked to find the probability that the combined total number of daily customers for both cafes is less than 500.

To solve this problem, we need to create a new random variable using the sum of the lambdas. This new random variable will represent the total daily customers for both cafes combined. With the combined lambda value, we can use our knowledge of the Poisson distribution to calculate the probability.

Step-by-step solution for Example 1

1. Assume independence: Before combining the random variables, it is important to assume that the time taken to clean each car is independent of each other.

2. Create a new random variable: Let's define a new random variable, T, which represents the total time taken to clean both cars.

3. Find the expected value: The expected value of the combined random variable is obtained by adding the means of the individual random variables.

4. Calculate the variance: To find the variance of the combination, we add the variances of the individual random variables.

5. Convert variance to standard deviation: Take the square root of the variance to obtain the standard deviation of the combination.

6. Use the normal distribution: With the mean and standard deviation of the combination, we can now use our knowledge of the normal distribution to calculate the desired probability.

Step-by-step solution for Example 2

1. Define independent random variables: In this case, the number of customers entering each cafe, X1 and X2, are considered independent random variables.

2. Identify lambda values: Determine the lambda value (expected number of daily customers) for each cafe.

3. Combine the random variables: Add the lambda values of X1 and X2 to create a new random variable, T, representing the total number of daily customers for both cafes.

4. Use the properties of the Poisson distribution: Recall the property that the sum of two independent Poisson random variables follows a Poisson distribution with the sum of their lambdas.

5. Calculate the desired probability: Utilize the Poisson cumulative distribution function (CDF) to find the probability of the combined total number of customers being less than a given value.

Practice questions in the RV Question Bank

To further enhance your understanding of combining random variables, it is recommended to practice more questions in the RV Question Bank. These questions will challenge you with different scenarios and complexities, ensuring a solid grasp of the concept.

Conclusion and further practice recommendations

In this article, we have explored the concept of combining random variables, the formulas for finding the expected value and variance, and illustrated their application through two examples - one involving a normal distribution and the other involving a Poisson distribution. To strengthen your skills in this area, Continue practicing with a variety of questions in the RV Question Bank. The more practice you get, the more confident you'll become in handling different combinations of random variables. Good luck!