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Solve the question related to the image: Let Y be John's total monthly income (including base and commission). Give the distribution of Y. (Hint: indicate the distribution of Y and find the mean and variance.)

To solve this question, we are given the following information:

- John’s base monthly salary is \( 5000 \) USD.
- His monthly commission \( X \) is normally distributed, \( X \sim N(4000, 500^2) \), meaning it has a mean of 4000 USD and a standard deviation of 500 USD.

### Let \( Y \) be John's total monthly income, which includes the base salary and commission:
\[ Y = 5000 + X \]

Since \( X \) is normally distributed and \( Y \) is just a linear transformation of \( X \), we know that \( Y \) will also be normally distributed.

### Distribution of \( Y \):
If \( X \sim N(\mu_X, \sigma_X^2) \), then \( Y = a + X \) follows the distribution:
\[ Y \sim N(\mu_Y, \sigma_Y^2) \]
where:
- \( \mu_Y = a + \mu_X \)
- \( \sigma_Y = \sigma_X \), since adding a constant does not affect the variance.

### Now, calculate the mean and variance of \( Y \):
1. **Mean of \( Y \):**
   \[
   \mu_Y = 5000 + 4000 = 9000 \text{ USD}
   \]
2. **Variance of \( Y \):**
   The variance remains the same as \( X \), so:
   \[
   \sigma_Y^2 = \sigma_X^2 = 500^2 = 250000 \text{ USD}^2
   \]

Thus, \( Y \), John’s total monthly income, follows the distribution:
\[
Y \sim N(9000, 500^2)
\]

This means John's total monthly income is normally distributed with a mean of 9000 USD and a standard deviation of 500 USD.

Would you like to explore more details or have further questions?

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Here are 5 related questions:
1. How do you calculate the probability that John's total income exceeds a certain value?
2. What is the probability that John's commission is between 3500 USD and 4500 USD?
3. How would John's income distribution change if his base salary increased by 500 USD?
4. How do we calculate the 95% confidence interval for John's total income?
5. How does the standard deviation of John's commission affect the spread of his total income?

**Tip:** When dealing with normal distributions, the mean shifts when constants are added, but the variance only changes with multiplicative factors.

Let Y be John's total monthly income (including base and commission). Give the distribution of Y. (Hint: indicate the distribution of Y and find the mean and variance.)

Learn how to determine the distribution of total monthly income when a base salary is added to a normally distributed commission. This problem covers key concepts of probability, normal distribution, and linear transformation of random variables. Suitable for Grades 11-12 or introductory college level.

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