Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 265

Fig. 4.19 Probability in Exercise 4.2

Fig. 4.20 Illustration of Exercise 4.4

Exercise 4.3 For a normal random noise X with a variance equal to 9 and a mean
value equal to zero, find the value c so that we find

j <c; (4.157)

99% of the time.

Answer Considering the mean value m ¼ 0 and the standard deviation s ¼ 3
while using ( 4.31 ) it follows:

3 p Pj <c f g ¼ Pfc<X<cg¼ 0 : 99 ¼ erf c= : (4.158)

Using erfinv.m we have:

erfinv ð 0 : 99 Þ¼ 1 : 8214 ; (4.159)

resulting in:

c ¼ 3 p 1 : 8214 ¼ 7 : 7275 : (4.160)

Exercise 4.4 A DC component U 0 is added to a normal noise with a mean value
and variance equal to 0, and s 2 , respectively. Next, the sum is amplified k -times, as
shown in Fig. 4.20a .
(a) Find the probability that the amplitude of the output noise Y will bemore than kU 0 .
(b) Determine whether or not this probability depends on the variance of the input
noise X .