Questions related to distributions Define Identically Distributed Define Indepen

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Questions related to distributions Define Identically Distributed Define Independent Distribution Define IID What does a delay of zero indicate? What would the maximum number of customers in a que represent? What does the busy function represent? why are systems normally not designed to run at 100% capacity? Interarrival and service times are modeled as independent random variables-why? Comment on the justification for 1 minute as the arrival time and .5 minute average service time with exponential density functions

Explanation / Answer

1) Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the sample are taken from the same probability distribution.

2) Independent Distributed means that the sample items are all independent events. In other words, they aren’t connected to each other in any way.

3) Two random variables X and are said to be i.i.d random variables ( i.i.d stands for independent and identically distributed) if and are independent r.v.s, and they have the same distribution.

Intuitively, and are independent if knowing the value/s of would give us no information that would change () for any , and vice versa. We state this formally as

(,)=()() for all ,.

The random variables and have the same distribution if ()=() for every .

For discrete r.v.s we can define i.i.d. r.v.s. equivalently as

As a concrete example, co4. nsider tossing two fair coins. Let be the indicator of heads on the first toss (thus, =1 if the first toss is heads, 0 otherwise), be the indicator of tails on the second toss (thus, =1 if the second toss if tails, 0 otherwise)

The r.v.s and are independent since (=,=)=(=)(=) )for all ,{0,1} ( I will leave this as an exercise).

Also, and have the same PMF since (=0)=(=0) and (=1)=(=1).

Note, though, we can have two r.v.s which are

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