Suppose that the amount of time (in months) that a light bulb lasts before it burns out has an exponential distribution with parameter λ = 1/5. a) what is the probability that it lasts at least three months? b) if it has already lasted two months, what is the probability that it will last at least three more months? c) on average, how many months will the light bulb last?

We have

a)

Thus probability that the bulb lasts 3 months equals

b)

It is a case of conditional probability

Let A be the event that the bulb will last another 3 months provided it has lasted for 2 months

'B' be the event that that the light bulb lasts for 2 months

thus we have

also

=

c)

On an average the expectation value of the life of the bulb will be given by mean of the distribution

thus on an average the bulb will last for 5 months.

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.