USAMA MAHMOOD MATH 205 Homework 3 Due: Thursday March 24 by 11.59 PM INSTRUCTIONS • Written part: Take pictures (or scan) the solutions of the written portion of the homework. Show all your work. If you wish, you may type your answers. • Computational part: Attach a screenshot of the R script and a screenshot of the output(s) of the script. • Combine the written and computational parts into a single pdf file. • To submit your work, go to the course canvas page > Assignments > HW3 and upload your solutions. ÷t¥¥ Written (1) Let X be a discrete random variable defined by p(x) = (a) Compute E(X) 8 > < x+1 ; x = 1, 2, 4 10 > :0; otherwise #⇐# Ecu)= Its)t2C%)tu(E) (c) Compute )+I(±)=¥ F-(2×5=2%-7+24,3 (2) Let X be a continuous random variable with a pdf given by 8 > <x; 0 x 2 2 p(x) = > :0; otherwise (a) Find the cdf F (x) of X. a) 700=2,1 (b) Compute P (0.5 X < 1). b) 7- (b) flag - 7- (2) -710.5) (c) Compute P (X > 1.5). 2 F- -10%5--0.1875 9--1=(1-5) c) = 1- = 4¥ 0.4375 1 -2£ vcxt-ECXY-CI-cxDZ-f-s-i-YE.lt/6C-) -(-#- - - (b) Compute V (X) E(2X ) = ¥ - %¥= -1¥ Computational (1) Consider the random variable defined in the written part, question 1. )={ 7cm (a) Simulate a data set of n = 500 random draws from the distribution. ¥ 0, in = 1,2M otherwise (b) Compute the mean of the data set. (c) Produce a bar plot of the data set. (d) Repeat part (a),(b),(c) with sample sizes n = 1000; 10000; 20000; 50000; 100000. Ég=h 0.2 Images in and R code is separate file . thankyou 2 . shared