MATH101 April 2005
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) •
Question 01 (g)
A continuous random variable is exponentially distributed with a mean of .
Find the probability that .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
You might begin by writing out the formula for the probability density function for an exponential random variable and express the probability that is greater than or equal to 8 in terms of an integral.
Think about how knowing the mean value may be helpful in terms of solving for any unknown parameters.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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We recall that the probability density function (PDF) for an exponential random variable is given by
The probability that is greater than or equal to 8 is given by integrating the PDF from to infinity.
Clearly, we need the value of if we are to evaluate the probability that is greater than or equal to 8. We can determine this by evaluating the mean of and comparing to the given value. The mean of is
Therefore the mean of is , and thus . Plugging this into the result above, we find that
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