Customers arrive at a service facility according to a Poisson process of rate λ customers/hour. Let X(t) be the number of customers that have arrived up to time t. Let W1,W2,... be the successive arrival times of the customers. (a) Determine the conditional mean E[W1|X(t)=2].
(b) Determine the conditional mean E[W3|X(t)=5].
(c) Determine the conditional probability density function for W2, given that X(t)=5.

Step-by-step explanation:

Given that:

X(t) = be the number of customers that have arrived up to time t.

... = the successive arrival times of the customers.

(a)

Then; we can Determine the conditional mean E[W1|X(t)=2] as follows;

Now

(b)  We can Determine the conditional mean E[W3|X(t)=5] as follows;

Now;

(c) Determine the conditional probability density function for W2, given that X(t)=5.

So ; the conditional probability density function of given that  X(t)=5 is:

r(0,90)(x,y) . t(-5,0)

last option

step-by-step explanation:

∆def to ∆d"e"f"

- rotate 90 counterclockwise degrees

- then translated 5 to the left (x --> x - 5)

answer is the last option