The water level of a certain reservoir is depleted at a...

The water level of a certain reservoir is depleted at a constant rate of 1000 units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson process with rate $.2$ per day. The amount of water added to the reservoir by a rainfall is 5000 units with probability $.8$ or 8000 units with probability .2. The present water level is just slightly below 5000 units. (a) What is the probability the reservoir will be empty after five days? (b) What is the probability the reservoir will be empty sometime within the next ten days?

The water level of a certain reservoir is depleted at a constant rate of 1000 units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson process with rate $.2$ per day. The amount of water added to the reservoir by a rainfall is 5000 units with probability $.8$ or 8000 units with probability .2. The present water level is just slightly below 5000 units.
(a) What is the probability the reservoir will be empty after five days?
(b) What is the probability the reservoir will be empty sometime within the next ten days?

Key Concepts

Poisson Process

A Poisson process is a stochastic model used to describe the occurrence of discrete events that happen independently and at a constant average rate over time. It is characterized by the property that the number of events in disjoint intervals are independent, and the number of events in an interval follows a Poisson distribution whose mean is proportional to the length of the interval.

Compound Poisson Process

A compound Poisson process extends the Poisson process by associating a random magnitude with each event. In such processes, the total effect over a time interval is the sum of random contributions from each event, where the number of events follows a Poisson distribution and each event’s magnitude is an independent random variable with a specified distribution.

Processes with Drift and Jumps

Many stochastic processes combine a deterministic trend, known as drift, with random jumps that occur at unpredictable times. The drift can represent a steady increase or decrease, while the jumps, often modeled using a Poisson or compound Poisson process, represent sudden changes. This combination is common in models where a system undergoes continuous change punctuated by abrupt, random events.

Ruin Theory in Risk Processes

Ruin theory is a branch of applied probability that studies the probability that a stochastic process, typically modeling an insurer’s surplus or a similar resource level, falls below a critical threshold. The theory involves assessing the balance between regular expenses or losses (modeled by drift) and random gains or claims (modeled by jump processes), aiming to calculate the likelihood of 'ruin' or depletion of the resource over a given period.

Transcript

00:01 Let's measure water in thousands of units so the numbers are simpler.

00:05 So the drain rate is 1 ,000 units per day.

00:07 Let's call that 1 in 1 ,000 units per day.

00:10 Then each rainfall adds 5 with a probably 0 .8 and 8 with the probability 0 .2.

00:15 The rainfall follows a poison process with a rate of 0 .2 per day.

00:20 So the present level is just under 5 ,000, which is going to be about 5.

00:25 So without rain, it empties at about day 5.

00:29 Let n of t be the number of rainfall is by time to.

00:32 N of t follows a poison distribution with a mean of 0 .2t.

00:41 In five days, the reservoir loses 5 ,000 units.

00:44 Since it starts just under 5, it'll be empty by day 5 unless a rainfall happens before then.

00:49 And any rainfall adds at least 5 ,000, which could help it not empty at day 5.

00:55 So the probability that we're empty at day 5 would be the probability that n of 5 is equal to 0.

01:06 So here, n of 5, follows a poison distribution of 0 .2 times 5, which is going to be a poison distribution with a mean of 1.

01:25 So, the probability that n of 5 is equal to 0 is going to be equal to e to the negative 1, which is about 0 .3679.

01:37 For part b, there are only two ways it can be empty by day 10.

01:41 Way 1 would be no rain in the first five days...

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