Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 273

Find the mean values of the sum and the product of variables X and Y .

Answer The mean value of the sum of random variables X and Y is:

EfX þ Yg¼EfXgþEfYg: (4.204)

From ( 4.203 ), it follows:

EfXg¼ 3 :

(4.205)

The mean value of the uniform variable Y is at the point of symmetry of its
PDF, i.e.,

EfYg¼ 0 : 5 : (4.206)

Placing ( 4.205 ) and ( 4.206 ) into ( 4.204 ), we get:

EfX þ Yg¼ 3 þ 0 : 5 ¼ 3 : 5 : (4.207)

The random variables X and Y are independent, and as a consequence we have:

EfXYg¼EfXgEfYg: (4.208)

Placing ( 4.205 ) and ( 4.206 ) into ( 4.208 ) we arrive at:

EfXYg¼ 1 : 5 : (4.209)

Exercise 4.16 N messages are transmitted over a channel. The times necessary to
transmit the messages are the independent variables X i with equal mean values
m i ¼ m and equal variances s i ¼ s 2 .
(a) Find the density of a random time in which all messages could be transmitted.
(b) Find the approximate total time needed to transmit all N messages.
Answer
(a) According to the CLT, the total time needed to transmit all messages is
approximately equal to the normal random variable

X ¼ X

N

X i Nðm X ; s X Þ; (4.210)

i¼ 1

where

m X ¼ X

X ¼ X

N N

s 2 s 2

i ¼ Ns 2

m i ¼ Nm ; :

(4.211)

i¼ 1 i¼ 1