# 205HW3 ```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.
• 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 &gt; Assignments &gt; HW3 and upload your
solutions.
&divide;t&yen;&yen;
Written
(1) Let X be a discrete random variable defined by
p(x) =
(a) Compute E(X)
8
&gt;
&lt; x+1 ; x = 1, 2, 4
10
&gt;
:0; otherwise
#⇐#
Ecu)= Its)t2C%)tu(E)
(c) Compute
)+I(&plusmn;)=&yen;
F-(2&times;5=2%-7+24,3
(2) Let X be a continuous random variable with a pdf given by
8
&gt;
&lt;x; 0  x  2
2
p(x) =
&gt;
:0; otherwise
(a) Find the cdf F (x) of X. a)
700=2,1
(b) Compute P (0.5  X &lt; 1). b) 7- (b) flag
-
7- (2) -710.5)
(c) Compute P (X &gt; 1.5).
2
F- -10%5--0.1875
9--1=(1-5)
c)
=
1-
=
4&yen;
0.4375
1
-2&pound;
vcxt-ECXY-CI-cxDZ-f-s-i-YE.lt/6C-) -(-#- - -
(b) Compute V (X)
E(2X )
=
&yen;
-
%&yen;= -1&yen;
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.
&yen;
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.
&Eacute;g=h
0.2
Images
in
and R code
is
separate file
.
thankyou
2
.
shared
```