Coxian Distribution

Coxian Distribution can be used to model:

Service time at a service center that provides bunch of service in a service with option of saying yes or no to continue at each stage.

The basic distribution of a Coxian distribution is

Exponential Distribution

and its pdf is given as

f(x) = μe^-μx ; x >= 0

and is generally used to model time span between 2 events that come poison distributed e.g. time between 2 calls in a call center, or time it takes to cut hair in barber shop.

Now, if 2 tasks where service-time/dwell-time are exponentially distributed are placed in

1. Series, they are called Hypo-exponential

e.g.

μ_1(hair-cut)   ---->    μ_2(shampoo) 

X1    ------>   X2

Now if we need to find sum of  2 random variable then we do convolution

pdf(X₁ + X2 = x) = ∫  μ₁e^-μ_1t * μ₂e^-μ_2(x-t) dt  

Hint:1.  t+(x-t) =  x; and for all possible value of "t" from -infinity to infinity

        2.  (x-t) in second exponent ranges for values greater than 0, and thus "t" is limited to values between 0 and x 

Hypo-exponential p(x) = ......................

∞−∞

2. Parallel, they are called Hyper-exponential

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Imagine there is a bag and there are 2 coins {A,B} inside it. An experimenter randomly picks a coin and tosses it to observe {Heads, Tails}.

So the pdf of observing Heads can be written as

P(outcome = Head) = (probability of selecting coin A)(probability of getting head in coin A) + (probability of selecting coin B)(probability of getting head in coin B)

}

to be continued