Question

Assume the length X, in minutes, of a particular type of telephone conversation is a random variable with probability density function

$gPNq7G+KgStuwAAAAASUVORK5CYII=$

(a) Determine the mean length E(X) of this type of telephone conversation.

(b) Find the variance and standard deviation of X.

(c) Find E[(X+4)2] $i2HyUzHDZnwAAAABJRU5ErkJggg==$ .

Answer 1

Sol Given that Assume the length x, in minutes of a particular type of telephone conversation is a gandom Vasiable with probability density function - 1/4x doo f(a) = ? a elsewhere (a) To determine the mean length E(x) of this type of telephone Conversation : - E(x) = (xf(x) dx -V ells de -xls da let 7 / = 2 x = 42 da = 4d2 and 5 270 7:0 1 x=0 => z=2 reduces to E(X) = 5 zé?(yd 2) E(X) = 4 (é? 22-'42 70

We know that • r(n = rent dx ,nxo (n+1) = no if nis integer From equation o Elx) = 54(2) E(x) = 4X!! F(x) = 4x1 JE(x) = 4 b)To find the variance and standard deviation of * F(x) = 5 axe dx -- *44 (3'48 (+4) ] = 32 E(x2) Therefore, V(X) = E (x2){[(x)7 = 32-(4) = 32-16 IV(x) = 16 50, the standard deviation = V V(*) VT6 = 4

08 = Get 8h = +x8 +93 +6€ = (*)28 +91+(x)2 = [x8+21x] = = E[(x+4)%] = [ *4+ 2x4 xx] () To find E [(x+4)*] :-