Question

le* 4. A random variable has a pdf f(x) = lo if x > 0 if xso , find the cdf, mean value and variance. Tel. :

Answer 1

f(x) = f ex x70 lo x=0 The CoF (Cumulative Distributive function) Fla) in area function of the PDF , obtained by integrating the PDF from -o to an arbitrary value x. (When a 70) x ft) dut flt) at t Todt t l ét dit o t et 1 t= x - = 'too 1 2 1 - é x | When x20 e F (2) [flt) dt dt = 0

WORDEN GORENTA CDF for. F(2) 7 0 T 1-ex 250 x70 - Notu x flada E(X) le Mean is equal to 01 Os E(X) = 120 d x + xexda ES 0 + l xex da .to ttco l Integrating by , part taking zas let in lem tros [-xe-xter] - . (0-0) - (0-1) = 1 everything about le long ban = of H i E(X) = 7

Var (X). EDO - LE «N) ] x²f(x) dx 0-0 1x² Oda + 1x² éxda - o x² ex dx - - 1 Now (x²ex dx x as 1st f" Integrating by parts $ taking Taxz ex - 22 ex dx Į (-1) (-1) -22 ? + 2(xet - x² * + 2 ( xexe + Leade axel - 2283-2007 Now [x²ex dx 0 -x ² -2 -2x e²_2ex

- 1 leto -2 -3 -10 +0-2). 0-0 - 0 +2 Verse (X) = 2-1-1