Embed Size (px)
Citation preview
@
. ~7.;:::: ~J(.1- % : 1.CJ0.
of,AuYl'1- ciAiJ,(lA,C')1,&t« u/ku (yyz+aj,1c., we- !eIl1J/lA) +tl'd r- c; 0/.u:(4
fa (J 9- II~C/.l ~ C<.LLtwed 'A.- value- dvvuLai h Lv0 It,~,.£903. ~CA- i.9b > 1~g03i u.: v~ i1 a/I/1 vu.1-0vs..
M!fifCuM,~ef k ~
11/) I~ <yVWJ v~ f ~ tA I 11~ <f61-tJ N c .\I) Ik ryu.uJ vafM-L r t~~ o(pv I a ~~-/6!1 c.
_ ) 1/~ {J :j,- 4.loP. ~ "'- f: '/L -t aI ').&J fu {I--i <JG/t1l) c L 0 @~
t:O
(I) J fu (x i -i14) J '}L
-cO--tOft). ~~J % ~*~i~~)-(i-*)J = i"i-'ct/~ ~ -\-t:11}'
ji) trJoo"t- fu (')/ .; / 4) oI1( ~'1 J f- J f)[
~~ -~-ct;)-+4/'1-
z: I ~ \~ :::: r}Z:r <1 i.- q/,}
~.e.{).
1;)
{ ~ a__ ~.e.D.
iii.
@
ICffmt~J ® ~@8').~ ("I -i ')C 1.
s . 0 6 Q.E.D
_ ® 1l'1eevvt f ?G z: i 'it PC il,;n ')n=u
A"l= I-- ('Yl~_,911,y;-t ry\ "-) p( 'l1 'l ,fl )()f\-,=O"" »o '"~ 2-- ry\'- p( rt1 i 11') ~ ~,yj 1'Yl P ( rr1 , -rt) --I- ,y; >- L f' ('"\ ,y; ')
'!'tOo 'h~u 'Yt~O
:0 ;r. (,y; -d_') - 'Y'\ L
.---~ ryl. 8·F-·D.
®. r: 'f) .:(1):=- 1. --11-~~ I ..--0 'YI111 I L-•
z. 31;iAtu;1J '(, ~: rYlL e: ~ 2,'(\0('1 Y1
~r(')1'\iVt) = ~(~C- \\ r(Ii\e:.i1V·7(.lI )
~ r{\ t1I ~ ') -\ ~'Y\-0) - fl-M ~
~.. K ~ ry1 .-~
__(k-t;;\) ~ (~\) -\ "" .- P-M ~lY-\.:{\1
\ l\<, "" 'Y) ) )
-~~~~-') ~
. (IV -\:,;;\') tv (3-- _\ ~ \'V - ~ ~ ?7\ (!:-\.v.). \~~~J -\ ~
~ ~ \« -\ ,vi,) ,\rYI( i -\1«/ 'Y'\1 A 1~ e-: ~ V·7\ ~~-\-0)
~._,~~~~)~\V K,/~)A\~-jM~)g[fV>LP~J frud ~ iWJ ~fAfdVl1( .
c -.JI /'<r l- /</ ; ~,gv _ g::;0 -+ ~?- -t- ;V e- ,em ~ .R'VI (!(--\rYi )
QM If) ~ ~ I\t '>- -Ie 1<::" tvL r- -'~ ~>-_ J~1'-- (je-I-'YI)
fov I"'f ~ 0-rn , ~ {l~ iF 1W- ~ ~.
~ K +0/\ ?b ;;;
17 (11) ~ f/Vlf (-~-Jf)~~
.-/. ~ ~ ry\"- ry'\
Fl~)~ ~t (-~~)~
~ R'1Cl1.~ leur ~ 0' rill ~ fot0071 of}:J tu'6~'0v!»:": o ~~ ~ lk '/ftL/t7VLr rJi/J ~cMfYl.
In[383]:= (*QUESTION 6*)
(*Mean*)
f = 1 2 Sin[x];
xbar = Integrate[f * x, {x, 0, Pi}] // N (* Mean *)
(* Variance *)
var = Integratef * x - xbar^2, {x, 0, Pi} // N
(*Skewness*)
s = FullSimplifyIntegratef * x - xbar^3, {x, 0, Pi} // N
(*Kurtosis*)
k = FullSimplifyIntegratef * x - xbar^4, {x, 0, Pi} // N
sigma = sqrt[var] // N;
(*for same mean and standard dev, skewness and kurtosis for the Normal Distribution*)
(*Skewness for Normal Distribution*)
S = Integratex - xbar^31
sigma * 2 PiExp
-x - xbar^2
2 * sigma^2, {x, 0, Pi} // N
(*Kurtosis for Normal Distribution*)
K = Integratex - xbar^(4)1
sigma * 2 PiExp
-x - xbar^2
2 * sigma^2, {x, 0, Pi} // N
Out[384]= 1.5708
Out[385]= 0.467401
Out[386]= 0.
Out[387]= 0.479255
Out[389]= 0.
Out[390]= 3. Erf 1.11072sqrt[0.467401]
sqrt[0.467401]4 +
2.71828- 1.2337
sqrt0.4674012 -3.09243 sqrt[0.467401] - 3.75994 sqrt[0.467401]3
(*While the skewness for both the curves is the same, zero,
the kurtosis for the Normal distribution is greater than that for the given function.*)
(*QUESTION 7*)
(*For N = 2 *)
list1 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 2, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list1, sum]; sum = 0;
]
Histogram[list1, 20]
(*For N = 8 *)
list2 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 8, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list2, sum]; sum = 0;
]
Histogram[list2, 20]
(*For N = 20 *)
list3 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 20, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list3, sum]; sum = 0;
]
Histogram[list3, 20]
(*For N = 50 *)
list4 = {}; sum = 0;
For[i = 1, i < 500, i++,
For[j = 1, j < 50, j++,
sum = sum + RandomVariate[UniformDistribution[{0, 1}]]
]
AppendTo[list4, sum]; sum = 0;
]
Histogram[list4, 20]
2 Assignment_Solution.nb
Out[365]=
0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
Out[368]=
2 3 4 50
10
20
30
40
50
Out[371]=
6 7 8 9 10 11 12 130
10
20
30
40
50
60
70
Assignment_Solution.nb 3
Out[374]=
20 22 24 26 28 300
10
20
30
40
50
4 Assignment_Solution.nb
Assignment_Solution.nb 5
