Examples (Random Signals)
Referring to Fig. B-2, we can show the mutually exclusive events A, B, C, D, E, F, G,
and H by a Venn diagram. These are all the possible outcomes of an experiment, so
the sure event is S = A + B + C + D + E + F + G + H. Each of these events is denoted
by some convenient value of the random variable x, as shown in the table in this
figure. The assigned values for x may be positive, negative, fractions, or integers as
long as they are real numbers. Since all the events are mutually exclusive,
P(S) = 1 = P(A) + P(B) + P(C) + P(D) + P(E) + P(F) + P(G) + P(H)
That is, the probabilities have to sum to unity (the probability of a sure event), as
shown in the table, and the individual probabilities have been given or measured as
shown. For example, P(C) = P (-1.5) = 0.2. These values for the probabilities may be
plotted as a function of the random variable x, as shown in the graph of P(x). This is a
discrete (or point) distribution since the random variable takes on only discrete (as
opposed to continuous) values.
Determine the CDF F(x) and PDF f(x) of x.
Note that the CDF start at a zero value on the left ( a  ) and the probability is
accumulated until the CDF is unity on the right (a  ) .
We obtain the PDF by taking the derivative of the CDF. The PDF consists of delta
functions located at the assigned (discrete) values of the random variable and having
weights to the probabilities of the associated events.
Continuous distributions:
Example
Let the random variable denote the voltages that are associated with a collection of a
large number of flashlight batteries (1.5-V cells). If the number of batteries in the
collection were infinite, the number of different voltage values (events) that we could
obtain would be infinite, so that the distributions (PDF and CDF) would be
continuous functions.
Suppose that, by measurement, the CDF is evaluated first by using F(a) = P(x - a),
The CDF that might be obtained is illustrated in Fig. B-5a, The associated PDF is
obtained by taking the derivative of the CDF, as shown in Fig. B-5b. Note that f (x)
exceeds unity for some values of x, but that the area under f (x) is unity (a PDF
property).
Suppose that we want to calculate the probability to obtaining a battery having a
voltage between 1.4 and 1.6.
1.6
p(1.4  x  1.6)   f ( x)dx  F (1.6)  F (1.4)  0.19
1.4
Tutorial questions (Random Signals)
Q.1
A standard deck of 52 playing cards is shuffled and two cards are drawn at random.
After looking at the first card, what is the probability that the second is a heart if
(a) the first card is a heart;
(b) the first card is not a heart?
(c) Calculate P(AB) for part (a).
Q.2
Consider the experiment that consists in the rolling of two honest dice. The random
variable X is assigned to the sum of the numbers showing up on the two dice.
Determine and plot the cumulative distribution function and the pdf of X.
Q.3
Consider the random variable x  cos( 0t1   ) , where θ is a random variable with a
uniform probability density over       . Compute the mean and rms value.
Q.4
The amplitude of a signal is Gaussian-distributed with zero mean and a mean-square
value σ2 . Find the probability of observing the signal amplitude above 3σ.
Q.5
A discrete random signal, y(t), can take one of the four predefined voltage levels,
y1 = 0.3 V, y2 = 0.2 V, y3 = 0.5 V, and y4 = 0.1 V. Assume that these levels occur
with probabilities, p(y1) = 0.2, p(y2) = 0.3, p(y3) = 0.1, and p(y4) = . Determine the
value for , and calculate the average power delivered by y(t) into a 50  resistive
load.
Q.6
A continuous random signal, x(t), is uniformly distributed over the range from
-2 V to +2 V. Determine the probability density function of x(t), and hence calculate
the average power delivered by x(t) into a 75  resistive load.