Compute E[X] if X has a density function given by a. f ( x ) = { 1 4 x e − x 2 x > 0 0 otherwise b. f ( x ) = { c ( 1 − x 2 ) − 1 < x < 1 0 otherwise c. f ( x ) = { 5 x 2 x > 5 0 x ≤ 5
Compute E[X] if X has a density function given by a. f ( x ) = { 1 4 x e − x 2 x > 0 0 otherwise b. f ( x ) = { c ( 1 − x 2 ) − 1 < x < 1 0 otherwise c. f ( x ) = { 5 x 2 x > 5 0 x ≤ 5
b.
f
(
x
)
=
{
c
(
1
−
x
2
)
−
1
<
x
<
1
0
otherwise
c.
f
(
x
)
=
{
5
x
2
x
>
5
0
x
≤
5
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
Q prove or disprove: If Ely/x) = x = c(dipy
=BCCo
(BVC)
ECxly)=y, and E(X2), Ely)
In a small office, there are m = 5 typists who need to use a single typewriter to complete their reports. Assume the time
each typist takes to prepare a report follows an exponential distribution with an average of 20 minutes per preparation
(A = 3 reports/hour), and the service time for the typewriter to type out a report also follows an exponential distribution,
averaging 30 minutes to complete a report (μ 2 reports/hour). Given that the number of typists is finite and all typists
=
share one typewriter, they will form a waiting queue.
(1). Describe this queuing system and explain how it fits the characteristics of the M/M/1/∞0/m model.
(2). Calculate the probability that any typist is using the typewriter at steady-state.
(3). Calculate the average number of typists waiting in the queue at steady-state.
(4). Considering the need to reduce waiting time, if an additional typewriter is introduced (turning into a two-server
system, or M/M/2/∞0/m model), analyze the expected impact,…
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