11-25. (Binomial distribution)
Forty identical transistors are placed on life tests simultaneously and are op-
erated for T hours. The probability that any transistor survives to time T is 0.8.
Let Xdenote a random variable that represents the number of transistors which are
operational at time T. The distribution function for Xis needed to compute proba-
bilities. The binomial distribution is applicable if (a) each transistor is identical and
has the same chance of failure as any other and (b) life testing of each transistor is
identical and is accomplished under separate independent conditions. Let success
mean survival of transistor to time T, then p= 0. 8 and n= 40. The probability
density function for the random variable Xis
f(x) =
(
40
x
)
(0 .8)x(0 .2)^40 −x, x = 0, 1 , 2 ,.. ., 40
(a) Find the probability that exactly 33 transistors are operational at time T.
(b) Find the probability that 3 transistors have failed by time T.
(c) Find the probability that at least 3 transistors have failed by time T (i.e. the
probability that no more than 37 have survived equals
∑^37
k=0
f(k))
(d) Find the probability that 80% of transistors survive.
11-26. (Poisson distribution)
Many random experiments involve time. In an experiment, at any instant of
time, either something happens or it does not happen and only one thing can occur.
The number of these things that happen in a prescribed time interval is observed
and recorded. The Poisson distribution describes such situations. Let events which
occur randomly in time be called random points. A random variable X will then
represent the number of random points that occur in the interval between times t= 0
and time t > 0. The probability of observing exactly xrandom points between 0 and
tis given by the probability density function
f(x) = f(x;t) =
(λt)xe−λt
x! , x = 0,^1 ,^2 ,...
where λt > 0 and λ > 0 represents the average number of random points per unit of
time. If λt = 9 and Xdenotes a random variable
(a) Show that f(x) =^9
xe− 9
x! and f(x+ 1) =
9
x+1 f(x)
(b) Find P(X > 4) i.e. 5 or more random points occur.
(c) Find P(X≤8) i.e. no more than 8 random points occur.
(d) Find P(8 ≤X≤12) i.e. between 8 and 12 random points occur.
