Page 83 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Consider the function given by

(0 otherwise
where k is a constant. Find the value of k such that p(x) can be the pmf of a discrete r.v. X.
Ans. k = 6/n2

It is known that the floppy disks produced by company A will be defective with probability 0.01. The
company sells the disks in packages of 10 and offers a guarantee of replacement that at most 1 of the 10
disks is defective. Find the probability that a package purchased will have to be replaced.
Ans. 0.004

Given that X is a Poisson r.v. and px(0) = 0.0498, compute E(X) and P(X 2 3).
Ans. E(X) = 3, P(X 2 3) = 0.5767

A digital transmission system has an error probability of per digit. Find the probability of three or
more errors in lo6 digits by using the Poisson distribution approximation.

Ans. 0.08
Show that the pmf px(x) of a Poisson r.v. X with parameter 1 satisfies the following recursion formula:

Hint: Use Eq. (2.40).
The continuous r.v. X has the pdf
- x2) 0 < x < 1
otherwise
where k is a constant. Find the value of k and the cdf of X
x10
Ans. k = 6; FX(x) =

The continuous r.v. X has the pdf
- x2) 0 < x < 2
otherwise
where k is a constant. Find the value of k and P(X > 1).
>
Ans. k=j;~(X 1)=
A r.v. X is defined by the cdf

(a) Find the value of k.
(b) Find the type of X.
(c) Find (i) P(4 < X I 1); (ii) P($ < X < 1); and (iii) P(X > 2).