Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 293

A.4.3. The normal PDF becomes a delta function with an area of 1 at the mean
value m of the random variable; thus indicating that a normal variable
becomes a deterministic value m with the probability of 1.
A.4.4. Both PDFs will have the same shape. The only difference is positions of the
PDFs on the x -axis. Each PDF is centered on its mean value.
A.4.5. Both PDFs are centered on the same mean value, but have different shapes.
A.4.6. The PDF of the sum of two independent random variables is equal to the
convolution of their PDFs.
On the other hand, knowing that the sum of independent normal random
variables is itself a normal random variable, it follows that a convolution of
two normal PDFs is also a normal PDF.
A.4.7. The characteristic function of a normal variable has the shape of a normal
PDF.
A.4.8. The probability of event A is:

¼ 0

0 : 5

2

PfAg¼Pjj 0 f : 5 g ¼ Pf 0 : 5 X 0 : 5 g¼ erf p : 3829 : (4.257)

The probability of event B is:

PfBg¼Pjj 0 f : 5 g ¼ Pf 0 : 5 X 0 : 5 g¼ 2 PfX 0 : 5 g

¼ 1 PfAg¼ 0 : 6171 : (4.258)

Therefore, event B is more probable.
A.4.9. It is not possible [KOM87, pp. 103-104]. However, the transformation,
known as a Box-Muller transformation, transforms a pair of independent
uniform variables into a pair of independent normal random variables
[MIL04, pp. 190-191], [KOM87, pp. 102-104], as described below.
The random variables X 1 and X 2 are both independent and uniform over
[0,1]:

1 for 0 <x 1 1 ;

f X 1 ðx 1 Þ¼ (4.259)

0 otherwise :

1 for 0 <x 2 1 ;

f X 2 ðx 2 Þ¼ (4.260)

0 otherwise :

Two new random variables Y 1 and Y 2 are obtained by the following
transformation:

p

2ln ðX 1 Þ

Y 1 ¼ cos ð 2 pX 2 Þ; (4.261)

p

2ln ðX 1 Þ Y 2 ¼ sin ð 2 pX 2 Þ: (4.262)