Question

Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2

Answer 1

TOPIC:Transformation of random variables.

Given that, Xom N (01) X₂ ~ N(OID Yindependently. MER. Now, the pdf of x, is- 67, (d) e So, the mgf of X, is - M4, C4) - E[et]. po e th. dx,(20)dre 24. VaR) e 3 (x+*+2+24+% Ź (x- 1212 - 27 T 11 (24-t²7 ae ? d4 +. 1 -J 3 () de 51 100 t19 L - where g(x) = (x =/et/ – pdf of n(t, 1).

Similarly, X2 ~N(0,1). Sun My, (t) = € (e tk). Define y = x - x - So, the mgf of y is - - My (t) = Elett) Due to independence = E = m 9) ml)

(5/2/2 ( 4 )/2 e 7. eth/ t/ - =e - which is the mgf of NOOID. By the uniqueness of mas; we get, • Y = (x1-x2) WN(01). 2) Defines W = y2 = ( x, x)/² Nowy the pdf of y is - dy (g) = - 1 e 2 ; JERI (9) = of of y =P (*. Here, - & Lyco. =) y 20. 2) w you

So, the cof of w is - > Fw (W) = p (wsw). =P (y2 300 = P(11 125) = P(VW SY EWW). * P/vw) - (-). So, the pch of w is - Sw (W) = - [ {w (w] o alta) - 91-10) 2 dy (ra) + aut. dy(reg. dw piele Le col La dw (W). out - i don, wyo. dom uso. Jawa ; otherwise,

-which is the required pdf.