Page 204 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 187

6.19 For Example 6.10, determine the jpmf of X 1 and X 2 . Determine the probability
that, of the 10 persons studied, fewer than 2 persons will be in the low-income
group and fewer than 3 persons will be in the middle-income group.
6.20 The following describes a simplified countdown procedure for launching 3 space
vehicles from 2 pads:
. Two vehicles are erected simultaneously on two pads and the countdown pro-
ceeds on one vehicle.
. When the countdown has been successfully completed on the first vehicle, the
countdown is initiated on the second vehicle, the following day.
. Simultaneously, the vacated pad is immediately cleaned and prepared for the
third vehicle. There is a (fixed) period of r days delay after the launching before
the same pad may be utilized for a second launch attempt (the turnaround time).
. After the third vehicle is erected on the vacated pad, the countdown procedure is
not initiated until the day after the second vehicle is launched.
. Each vehicle is independent of, and identical to, the others. On any single
countdown attempt there is a probability p of a successful completion and a
probability q (q ˆ 1 p) of failure. Any failure results in the termination of that
countdown attempt and a new attempt is made the following day. That is, any
failure leads to a one-day delay. It is assumed that a successful countdown
attempt can be completed in one day.
. The failure to complete a countdown does not affect subsequent attempts in any way;
that is, the trials are independent from day to day as well as from vehicle to vehicle.
Let X be the number of days until the third successful countdown. Show that the
pmf of X is given by:

k r†!
2 k r 2
3 k 3
p k†ˆ k r 1†p q 1 q r 1 †‡ p q ; k ˆ r ‡ 2; r ‡ 3; .. . :
X
2 k r 2†!
6.21 Derive the variance of a Poisson-distributed random variable X as given by
Equation (6.47).
6.22 Show that, for the Poisson distribution, p (0, t) increases monotonically and then
k
decreases monotonically as k increases, reaching its maximum when k is the largest
integer not exceeding t.
6.23 At a certain plant, accidents have been occurring at an average rate of 1 every 2
months. Assume that the accidents occur independently.
(a) What is the average number of accidents per year?
(b) What is the probability of there being no accidents in a given month?
6.24 Assume that the number of traffic accidents in New York State during a 4-day
memorial day weekend is Poisson-distributed with parameter ˆ 3.25 per day. Deter-
mine the probability that the number of accidents is less than 10 in this 4-day period.
6.25 A radioactive source is observed during 7 time intervals, each interval being 10
seconds in duration. The number of particles emitted during each period is
counted. Suppose that the number of particles emitted, say X, during each
observed period has an average rate of 0.5 particles per second.
(a) What is the probability that 4 or more particles are emitted in each interval?
(b) What is the probability that in at least 1 of 7 time intervals, 4 or more particles
are emitted?

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