Question

3. Let X.. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X,. Sketch the pdf fx. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it can be shown that the pdf of Z is fz(2) = Le if :< -3 or 2 > 3 1 (2+3)2 if2 € (-3,-1] -1,2 ifz € (-1,1] 2(2-3)2 if : € (1,3) Note that you do not need to show this yourself. Carefully draw a graph of this pdf - you may use R to do so if you wish. Using fz, or otherwise, compute the expected value and variance of Z. Compute the values of fz(x) and (2) at z €{-3, -2,-1,0} to two leading decimals, where (2) = 7e-24/2 is the density of a standard normal, and add a sketch of o to your graph. Comment. Hint: you may wish to mention an important theorem in your comment. 5]

Answer 1

Short answer

S = X1+ X2+ ..Xn

where Xi = Unif(0,1)

E(S) = n/2, Var(S) = n/12

put n = 1,2,3 for a),b) and c) part

Long answer

ANSWER! Given that A let X; id ul-1, 1) for it f 1, 2, 3} a) The expected value and variance of x, (X), - '+1.1) zo VOX (1+1)? > pdf of x, is given by a fx, (xi) - , it 12X, 31 o, ow (0:1

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0 -2<U <2, -12 V 21 * from @ -12V <1; -1<U-V <1 - - - U - V < 1-U U-12V <1tu -> NOW, -22 U L-12- 1vL ltu -1<ulo ->-1<ULI tu for Olues ou leuci -1 1- 6 1V Scanned with

= for kuca 2 u 1 <UL a) The joint pdf of (u, v) is given by fluive) - Yu , (range as given above). I flu, v) so, when - < u2o. J f (0,0) do, when OLUK2 I /u (244) when -aculo Yu (asu), when oluce i pot of y = x, 4X, is given by. Yu (ary) when ngay co (/(2-y), when ocyer » ty (y) = f u lary) when

12:30 y 100) >> Ely) - Ya 3°04 + y) dy + yu[(97-99) ay - Ya [ye + 9%]* + Ya [gº. 6%). > Yu Cor (un 3/3) + (4-8%)-o] => € (94) - Yuſ (93499 89 1 ya] ing-g3ly • Yu [%]++ 9% ); Va[92 92-9%) - Yu [o- (- / + 1% % ) + ( 1% -1%2] - / -> vly) - 6099-7*(Y) = 16 CS Scanned

8,129. 1 o, if 23-3 of 223 I 118 (2+3), if 26 (-3,-1) | 3/8 / 2², if ZE (-1, 1) ( 16 (2-3)? , if Ze (1,3) (2) (-3,0) (0) (1,0) 2 -> > €0) - Ti '2 (1? +62 +9) dt + 51 * 'gell de 167 + 3) dzt + [ & 2 (2-3) 12 - You f'(2's re'se) et 13 (32-6,0) & J 1/16 (2? + 32 - 62 ? ) dz

> + n°4 25 x]+(25 %;- *%) = "ht [ 2% + 50; – 220] Me [[Xy - 2 4 %; )+ (%-54* 9) +1 % +7,7-54)(+3-)}, -> 429-% % (7+46° +5°) +5(2,22 keybe z facol pred + ', $(2472432) de fz12) (2) 0.00 uu32 0.0625 0.0539a1 0.25 0.241971 0.375 0.398 uur 2 > * The distribution of '2" is tending to standard normal distribution. Scanned with