Question

(Mathematical statistics)

* If
Z1
, $Z_{2}$ are independent standard normal random variables,

find the density of $\frac{Z_{1}}{Z_{2}}.$

Answer 1

i Given. 21 22 be independent standard normal (21) -L2, co f (21) = 21 -BL2ILOS 2n [ A(z) - e z ? - <Z2 LA 27 joind density of 2, f22 is P(2,1 2) = f(21) f (2) ... (2. 622 are inclependent) 22 √2T f (2,122) = ₂ (Zi²+ 22 ) 12 2T consider transformation, Y = 21 & Y2 = 22 o The 21 = 4.12 4 22 242. rang of y, 442 are - 242 Loo. میرے دل find socobian 15142 Joint dencity of (1,42) 14.142) = f (2, 2)/2 = 442, 2. J! 21 42 -(2, +2, 2 2124, 12 22 = 42

f(YY) = -(4,²4² +42 ²/2 27 -OLY, LOS fluu42) o e-42 (41²+1) 12 -OLY2LOO 2T 22 We have to find dencity of 21 - 4 - f(1) = f( 21 ) = 8 Y 2 e 72 (41²+1)/2 dyz 20 42=.0 = y2e72 (41"+1)/2 dl = + 2 y 2 e 72 (4171)/2 Dut 42 (48 +1) at 242dY2 = at . 42+1

RIA) = + en ot o acould ; -olyi Lao 77 (4+) Cauchy distribution, which is odencity of standard i e fly = f(21) occoil).