1. The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter λ. However, such a random variable can only be observed if it is positive, since if it is 0 then we cannot know that such an insect was on the leaf. If we let Y denote the observed number of eggs, then

P{Y = i} = P{X = i|X > 0}

where X is Poisson with parameter λ. Find E[Y].

2. The amount of time that a certain type of component functions before failing is a random variable with probability density function

f(x) = 2x, 0 < x < 1

Once the component fails it is immediately replaced by another one of the same type. If we let Xi denote the lifetime of the i-th component to be put in use, then Sn = Σ(i=1, n) Xi represents the time of the n-th failure. The long-term rate at which failures occur, call it r, is defined by

r = lim (n/Sn)

Assuming that the random variables Xi, i >or= 1 are independent, determine r.

Any help will be appreciated.

P{Y = i} = P{X = i|X > 0}

where X is Poisson with parameter λ. Find E[Y].

2. The amount of time that a certain type of component functions before failing is a random variable with probability density function

f(x) = 2x, 0 < x < 1

Once the component fails it is immediately replaced by another one of the same type. If we let Xi denote the lifetime of the i-th component to be put in use, then Sn = Σ(i=1, n) Xi represents the time of the n-th failure. The long-term rate at which failures occur, call it r, is defined by

r = lim (n/Sn)

Assuming that the random variables Xi, i >or= 1 are independent, determine r.

Any help will be appreciated.

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