Mathematical and Computational Methods for Engineers
E155C, Winter 2004
Handout #2
Discrete and Continuous Distributions, Moments, Distributions of Several Variables
Distributions, Moments, Change of Variables
1. Suppose that random variable X is the number that a
fair die turns up
a. sketch the probability density function f(x)
b. sketch the cumulative distribution function F(x)
2. Let  denote a continuous random variable
corresponding to the angular position of a pointer that is
free to spin about its center such that 0    2 .
a. determine f ( )
b. determine F ( )
c. determine P(0     / 3)
3. In tossing a fair coin, let X denote the number of trials
until the first head appears.
a. determine f ( x j )
b. show that

f (x j )  1
xj
c. what is the probability that the first head appears
in the first 3 trials ?
4. In the case of the spinning arrow determine the
following:
a. mean
b. variance
c. standard deviation
5. In producing ball bearings, the manufacturing process
has shown a radius variation of  2  0.09 mm. Using
Chebyshev’s inequality, estimate the upper bound on
the probability that the radius will fall more than 0.9
mm from the mean.
6. Find the output density for Y if X is a random variable
with PDF f (x) and y  x3 .
Binominal and Poisson Distributions
7. A production process is partitioned into two
independent sub-processes. The probabilities of a
defective component in the first and second subprocesses are 0.01 and 0.02, respectively. If 50 units are
produced, what is the probability there will be less than
2 defective units ?
8. Communication channels do not always transmit the
correct signal. Suppose that for a particular channel the
error rate is 1 incorrect transmission per 100 messages.
If 200 messages are sent in a given week and it is
assumed that their transmissions are independent, what
is the probability there will be at least 3 errors ?
9. It has been determined that 90 vehicles per minute
arrive at an intersection. Suppose X is the number of
arrivals in a 6-second interval.
a. find the density for X
b. find the probability that exactly 2 cars arrive at
the intersection within a 6-second interval
Normal Distribution
10. In a digital communication system, data is represented
by electrical signals. Let bits 0 and 1 be represented by
2 and 3 volts, respectively. The signal is distorted by
noise modeled as a normally distributed random
variable with   0 and   0.22 . The terminal
recognizes bit 0 if V  2.6 volts and bit 1 if
V  2.6 volts. What is the probability that the signal is
recognized incorrectly ?
11. To check a person’s telepathic ability, a screen is placed
between the investigator and the subject, and the
subject is asked to choose the one among five cards to
which the investigator is pointing. If there are 200
trials, what is the probability that the subject will get
more than 55 correct responses ? Use normal
approximation to the binomial distribution.