**Problem Statement:**Ben purchases a small life insurance policy that provides a payment if he dies within three years. The details are as follows:- If Ben dies within one year, the insurance payment is $5000.- If he dies in the second year, the insurance payment is $4000.- If he dies in the third year, the insurance payment is $2500.- There is no insurance payment if Ben survives the three-year period.Assume that if Ben is alive at a given point in time, the probability that he will survive the next one-year period is 0.85.Let $ X $ denote the insurance payment from the policy.**Task:**(a) Show that the values of the probability mass function (pmf) $ f(x) $ of $ X $ are as given in the table below.*Note: The problem requires computing or verifying the probability mass function for the insurance payments based on Ben's survival probabilities.* The image contains a table and a question related to the computation of moments. The table lists values of $ x $ and corresponding values of $ f(x) $.\[\begin{array}{c|c}x & f(x) \\\hline0 & 0.614125 \\2500 & 0.108375 \\4000 & 0.1275 \\5000 & 0.15 \\\end{array}\]Below the table, there is an instruction:(b) Compute the 2nd moment of $ X $ about the point $ b = 2500 $.**Explanation of the Task:** The second moment about a point (also known as the second central moment) is a measure used in statistics to understand the variance or spread of a set of values in relation to a specified point. In this case, the specified point is $ b = 2500 $. To compute this, use the formula:\[M_2 = \sum [(x_i - b)^2 \cdot f(x_i)]\]where $ x_i $ are the values of $ x $, $ f(x_i) $ are the corresponding function values, and $ b = 2500 $.

4. Ben purchases a small life insurance policy that provides a payment if he dies within three years. More specifically, if Ben dies within one year, the insurance payment is $5000. If he dies in the second year, the insurance payment is $4000. If he dies in the third year, the insurance payment is $2500. There is no insurance payment if Ben survives the three year period. Assume that if Ben is alive at a given point in time, then the probability he will survive the next one year period of time is 0.85. Let X denote the insurance payment from the policy.

4. Ben purchases a small life insurance policy that provides a payment if he dies within three years. More specifically, if Ben dies within one year, the insurance payment is $5000. If he dies in the second year, the insurance payment is $4000. If he dies in the third year, the insurance payment is $2500. There is no insurance payment if Ben survives the three year period. Assume that if Ben is alive at a given point in time, then the probability he will survive the next one year period of time is 0.85. Let X denote the insurance payment from the policy.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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