Page 120 - Applied statistics and probability for engineers

98 Chapter 3/Discrete Random Variables and Probability Distributions

(a) Probability that exactly 10% of the patients in the sample The probability that at least one site has lesions present is
fail to adhere. greater than or equal to 0.9.
(b) Probability that fewer than 10% of the patients in the sample 3-156. A utility company might offer electrical rates based
fail to adhere. on time-of-day consumption to decrease the peak demand in
(c) Probability that more than 10% of the patients in the sample a day. Enough customers need to accept the plan for it to be
fail to adhere. successful. Suppose that among 50 major customers, 15 would
(d) Mean and variance of the number of patients in the sample accept the plan. The utility selects 10 major customers ran-
who fail to adhere. domly (without replacement) to contact and promote the plan.
3-155. Suppose that lesions are present at 5 sites among 50 in a (a) What is the probability that exactly two of the selected
patient. A biopsy selects 8 sites randomly (without replacement). major customers accept the plan?
(a) What is the probability that lesions are present in at least (b) What is the probability that at least one of the selected
one selected site? major customers accepts the plan?
(b) What is the probability that lesions are present in two or (c) Instead of 15 customers, what is the minimum number of major
more selected sites? customers that would need to accept the plan to meet the fol-
(c) Instead of eight sites, what is the minimum number of sites lowing objective? The probability that at least 1 selected major
that need to be selected to meet the following objective? customer accepts the plan is greater than or equal to 0.95.

3-9 Poisson Distribution

A widely-used distribution emerges from the concept that events occur randomly in an inter-
val (or, more generally, in a region). The random variable of interest is the count of events that
occur within the interval. Consider the following example.

Example 3-30 Wire Flaws Flaws occur at random along the length of a thin copper wire. Let X denote the ran-
dom variable that counts the number of laws in a length of T millimeters of wire and suppose that

the average number of laws per millimeter is λ.
T
We expect E X) = λ from the dei nition of λ. The probability distribution of X is determined as follows. Partition
(
the length of wire into n subintervals of small length Δt = T n (say, one micrometer each). If the subintervals are chosen
/
small enough, the probability that more than one law occurs in a subinterval is negligible. Furthermore, we can inter-

pret the assumption that laws occur at random to imply that every subinterval has the same probability of containing a

l aw, say p. Also, the occurrence of a law in a subinterval is assumed to be independent of laws in other subintervals.

Then we can model the distribution of X as approximately a binomial random variable. Each subinterval generates
an event (law) or not. Therefore,

E X) = λ T = np
(
and one can solve for p to obtain
p = λ T
From the approximate binomial distribution n
⎛ n⎞
−
P X = x) ≈ ⎜ ⎟ p ( − p) n x
x
1
(
x⎠
⎝
With small enough subintervals, n is large and p is small. Basic properties of limits can be used to show that as n
increases
⎛ n⎞ ⎛ λ T ⎞ x λ ( T) x ⎛ λ T ⎞ − x ⎛ λ T ⎞ n −λ T
⎜ ⎟ ⎜ n ⎠ ⎟ → ⎜ ⎝ 1 − n ⎠ ⎟ → 1 ⎜ ⎝ 1 − n ⎠ ⎟ → e
x⎠ ⎝
⎝
Therefore, x!
λ
lim (X = ) x = e −λ T ( T ) x , x = 0 , , ,…1 2
P
n→∞ ! x
Because the number of subintervals tends to ini nity, the range of X (the number of l aws) can equal any nonnegative
integer.