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In a new MOBA video game, the damage dealt by a character is a random variable, X, with cumulative distribution function (x-1)² for 1 10 a) Find the probability density function of X. b) Find the mean and variance of X. c) After buying and equipping the “Hextech Gauss Bow" , the damage dealt by the character is transformed to the random variable Y = 2X + 5. Find the probability density function of Y.

In a new MOBA video game, the damage dealt by a character is a random variable, X, with cumulative distribution function (x-1)² for 1 10 a) Find the probability density function of X. b) Find the mean and variance of X. c) After buying and equipping the “Hextech Gauss Bow" , the damage dealt by the character is transformed to the random variable Y = 2X + 5. Find the probability density function of Y.

A First Course in Probability (10th Edition)
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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In a new MOBA video game, the damage dealt by a character is a random variable, X, with cumulative distribution function (x-1)² for 1<x < 10; 81 F(x) = for x < 1 1 for x > 10 a) Find the probability density function of X. b) Find the mean and variance of X. c) After buying and equipping the "Hextech Gauss Bow" , the damage dealt by the character is transformed to the random variable Y = 2X +5. Find the probability density function of Y. d) Using R, plot the probability density functions of X and Y on the same plot. Discuss the differences between the distributions, in terms of range, mean and variance. e) Using R: i. By applying an appropriate transform of a uniform random variable, sample 50,000 observations from X and produce a histogram of their distribution. Make sure to set your seed to your student ID before proceeding. ii. multiply each of your 50,000 observations in i. by 2 and add 5, and produce a histogram of the transformed observations. iii. compare your histograms from i. and ii. with the probability distribution functions plotted in d).
Transcribed Image Text:In a new MOBA video game, the damage dealt by a character is a random variable, X, with cumulative distribution function (x-1)² for 1<x < 10; 81 F(x) = for x < 1 1 for x > 10 a) Find the probability density function of X. b) Find the mean and variance of X. c) After buying and equipping the "Hextech Gauss Bow" , the damage dealt by the character is transformed to the random variable Y = 2X +5. Find the probability density function of Y. d) Using R, plot the probability density functions of X and Y on the same plot. Discuss the differences between the distributions, in terms of range, mean and variance. e) Using R: i. By applying an appropriate transform of a uniform random variable, sample 50,000 observations from X and produce a histogram of their distribution. Make sure to set your seed to your student ID before proceeding. ii. multiply each of your 50,000 observations in i. by 2 and add 5, and produce a histogram of the transformed observations. iii. compare your histograms from i. and ii. with the probability distribution functions plotted in d).
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