Question

please answer question 22

* 33 334 22. Let Xi and X, are continuous random variable with densities f(x) = 1 SIS2 and (0, Otherwise 9(3) 22 a respectively, where a, b > 0 are constants. 10, Otherwise (i) Find the cumulative distribution function Fx:(t) of X. (ll) Find the cumulative distribution function Fx.(t) of Xy. Your answer may involve a, b. (iii) Find the 50th percentile of X,. (iv) Find a ondo such that X; and Xhave the same 50th percentile. 1 l

Answer 1

are continuous random variable with x, and 2 densities fro) = 14 3<xsa x12 12xc a and 0iwi g(x) az a respectively o. IjFxi(t) a Ex, (t) = fox) dx -00

for XL F(t) = foda 30 for 12x42 Ele) e Jode of de f(t) = 4(2-. for 35x24, Fle) e Jotkują, de de = 4x3 + 4 |t-3) Elt) = t for 4LXLO

F(t) = lo dx & fa da t f 1 dx + f o da 0 SO -LILO F(x) = (2-1) 1 LIL2 080 4 3s RLG x 3, 4 Similasty, for gx) as g(x ) = 12,9 0.1 for a car ct) to do For xca f (+ to for x7, a F(t) = f fire) de t ffery de -ot ( b doo

F(t) = b 1 0 ] | 5 11 | frt) = L L 1 2 Q2. SO | F(X) ) F(x) =) 일 <a 1 (ex) → 229 | x

50th percentile : FCM) - 1 es f(2) = 4-1 = 9 4 f(3) = 3 F(4)= 1 m2-1 m2_1 = 2 m2=3 me 123 Im z 1.732 1 - 13 is not possible, it goes out of range 2 * for X2: F(m) = 1

2 و اه و [ 1 و یه 1232 = m ووو .2 2.999 - 2 ء 999-2 and g(x) dx = اعه و - 2 - 3 ع [ - و اع

292 using and ③ we get 8 292 2.ggg = 2a²= 2.999 2 = 1.4995 la = 1.2245 l so, b=2a² b = 2.ggg