Let X be the total medical expenses (in 1000s of dollars) incurred by a particular individual during a given year. Although X is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf fx) = 1+ for x 2 0. (a) What is the value of k? 1.8 (b) Graph the pdf of X. f(x) f(x) fox) fox) 1.0 1.0 1.0 1.0 0.8 08 0.8 0.8 0.6 0.6 0.6 0.6 0.4 04 0.4 04 02 0.2 0.2 0.2 (c) What is the expected value of total medical expenses? (Round your answer to the nearest cent.) s(0.833 Ox What is the standard deviation of total medical expenses? (Round your answer to the nearest cent.) $208 (d) This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (incle deductible amount) is $2500. Let Y denote the amount of this individual's medical expenses paid by the insurance company. What is the expected value of Y? [Hint: First figure out what value of X corresponds to the maximum out-of-pocket expense of $2500. Then write an expression for Yas a function of X (which involves several different pieces) and calculate the expected value of this function.] (Round your answer to the nearest cent.) $ 10.500
Let X be the total medical expenses (in 1000s of dollars) incurred by a particular individual during a given year. Although X is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf fx) = 1+ for x 2 0. (a) What is the value of k? 1.8 (b) Graph the pdf of X. f(x) f(x) fox) fox) 1.0 1.0 1.0 1.0 0.8 08 0.8 0.8 0.6 0.6 0.6 0.6 0.4 04 0.4 04 02 0.2 0.2 0.2 (c) What is the expected value of total medical expenses? (Round your answer to the nearest cent.) s(0.833 Ox What is the standard deviation of total medical expenses? (Round your answer to the nearest cent.) $208 (d) This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (incle deductible amount) is $2500. Let Y denote the amount of this individual's medical expenses paid by the insurance company. What is the expected value of Y? [Hint: First figure out what value of X corresponds to the maximum out-of-pocket expense of $2500. Then write an expression for Yas a function of X (which involves several different pieces) and calculate the expected value of this function.] (Round your answer to the nearest cent.) $ 10.500
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Question
![**Title:** Understanding Medical Expenses Distribution
**Text:**
Let \( X \) be the total medical expenses (in thousands of dollars) incurred by a particular individual during a given year. Although \( X \) is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with the probability density function (pdf) \( f(x) = k \left( 1 + \frac{x}{2.5} \right)^{-3} \) for \( x \geq 0 \).
**(a)** What is the value of \( k \)?
- Answer: \( 1.8 \) [Note: Incorrect solution]
**(b)** Graph the pdf of \( X \).
**Graph Explanation:**
There are four graphs displayed:
1. **First Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- The curve represents a decreasing function that starts at the top of the Y-axis and declines towards the X-axis as it moves right.
2. **Second Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Similar to the first graph but is incorrect as indicated by a red cross mark.
3. **Third Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Shows an increasing curve, incorrect as indicated by a red cross mark.
4. **Fourth Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Depicts an increasing function, incorrect as indicated by a red cross mark.
- The second graph is marked as correct with a check mark.
**(c)** What is the expected value of total medical expenses? (Round your answer to the nearest cent.)
- Answer: $0.833 [Note: Incorrect solution]
What is the standard deviation of total medical expenses? (Round your answer to the nearest cent.)
- Answer: $2.08 [Note: Incorrect solution]
**(d)** This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (including the deductible amount) is $2500. Let \( Y \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdbb6667-2d1e-4df4-b920-eeb3e502e5d9%2F71bd646f-6d69-4080-afeb-69a8c6cd19cf%2F3aclqfa_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Understanding Medical Expenses Distribution
**Text:**
Let \( X \) be the total medical expenses (in thousands of dollars) incurred by a particular individual during a given year. Although \( X \) is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with the probability density function (pdf) \( f(x) = k \left( 1 + \frac{x}{2.5} \right)^{-3} \) for \( x \geq 0 \).
**(a)** What is the value of \( k \)?
- Answer: \( 1.8 \) [Note: Incorrect solution]
**(b)** Graph the pdf of \( X \).
**Graph Explanation:**
There are four graphs displayed:
1. **First Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- The curve represents a decreasing function that starts at the top of the Y-axis and declines towards the X-axis as it moves right.
2. **Second Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Similar to the first graph but is incorrect as indicated by a red cross mark.
3. **Third Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Shows an increasing curve, incorrect as indicated by a red cross mark.
4. **Fourth Graph:**
- X-axis from 0 to 8, Y-axis from 0 to 1.
- Depicts an increasing function, incorrect as indicated by a red cross mark.
- The second graph is marked as correct with a check mark.
**(c)** What is the expected value of total medical expenses? (Round your answer to the nearest cent.)
- Answer: $0.833 [Note: Incorrect solution]
What is the standard deviation of total medical expenses? (Round your answer to the nearest cent.)
- Answer: $2.08 [Note: Incorrect solution]
**(d)** This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (including the deductible amount) is $2500. Let \( Y \)
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