According to the article “Reliability Evaluation ofHard Disk Drive Failures Based on CountingProcesses” (Reliability Engr. and System Safety, 2013:110–118), particles accumulating on a disk drive comefrom two sources, one external and the other internal.The article proposed a model in which the internalsource contains a number of loose particles W having aPoisson distribution with mean value m; when a looseparticle releases, it immediately enters the drive, and therelease times are independent and identically distributedwith cumulative distribution function G(t). Let X denotethe number of loose particles not yet released at a particulartime t. Show that X has a Poisson distribution withparameter m[1 – G(t)]. [Hint: Let Y denote the number ofparticles accumulated on the drive from the internalsource by time t so that X 1 Y 5 W. Obtain an expressionfor P(X 5 x, Y 5 y) and then sum over y.]
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
According to the article “Reliability Evaluation of
Hard Disk Drive Failures Based on Counting
Processes” (Reliability Engr. and System Safety, 2013:
110–118), particles accumulating on a disk drive come
from two sources, one external and the other internal.
The article proposed a model in which the internal
source contains a number of loose particles W having a
Poisson distribution with mean value m; when a loose
particle releases, it immediately enters the drive, and the
release times are independent and identically distributed
with cumulative distribution
the number of loose particles not yet released at a particular
time t. Show that X has a Poisson distribution with
parameter m[1 – G(t)]. [Hint: Let Y denote the number of
particles accumulated on the drive from the internal
source by time t so that X 1 Y 5 W. Obtain an expression
for P(X 5 x, Y 5 y) and then sum over y.]
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