Poisson Distrobution-HELP

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In summary: Good luck!In summary, the problem involves finding the probability of a breakdown occurring during an 8-hour workday for a robot with an average of 0.5 breakdowns per day. This can be calculated using the Poisson Distribution formula, where the rate (lamda) is equal to 0.5. The probability of a breakdown occurring during the day is approximately 0.6065, and the probability of the robot working for at least 4 hours without a breakdown is approximately 0.2865. The previous day's events do not affect the probability of a breakdown occurring.
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akito458
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Poisson Distrobution-HELP! URGENT!

the problem:

The Breakdowns of a robot follow a Poisson Dist. with an avg of .5 breakdowns per 8-hour workday. If this robot is placed in service at the beginning of the day, find the probability that:

a. It will break down durring the day.
b. It will work for at least 4 hours without breakdown
c. Does what happened the day before have any effect on your answers? why?

I'm a newb to probability and am I'm haveing trouble figuring out where to begin. I know that the rate(lamda) has something to do with the percent of break downs and the work day. Is it .5*workday or just .5. I just don't know where to begin with this problem. :cry:
 
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  • #2
i know that P{X >= a } = e^(-גa) where ג is lamda, would P{ X < a } = -e^(-גa)?
 
  • #3
Well the answer to a. is

P{ X < a } = e^(-גa)
= e^(-0.5)
= 0.6065

where ג = 0.5 <- the rate equals lamda

Since P{ X >= a } = 1 - ( 1 - e^( גa ) ) = e^(-גa)

right?
 
  • #4
i need help on b
 
  • #5
anybody...
 
  • #6
Answer

akito458 said:
the problem:

The Breakdowns of a robot follow a Poisson Dist. with an avg of .5 breakdowns per 8-hour workday. If this robot is placed in service at the beginning of the day, find the probability that:

a. It will break down durring the day.
b. It will work for at least 4 hours without breakdown
c. Does what happened the day before have any effect on your answers? why?

I'm a newb to probability and am I'm haveing trouble figuring out where to begin. I know that the rate(lamda) has something to do with the percent of break downs and the work day. Is it .5*workday or just .5. I just don't know where to begin with this problem. :cry:


A Poisson Process is a counting process; in your case define a success or occurrence when a breakdown happends. The probability of having n breakdowns within a period of time t is: P{N(t)=n}=exp(-lamda*t)*(lamda*t)^n/n!
In this case lamda = 0.5 breakdowns/day (since a day has 8 hours of work. Also, we can express lamda per hour as lamda = 0.5/8 = 1/16 breakdowns/hour.

a. It will break down during the day.
Here we can look for the probability of having a breakdown in a day (8 hours period), then P{N(1) >= 1} = 1 - P{N(1) < 1} = 1 - P{N(1) = 0} = 1 - exp(-0.5)
Or if we look for the probability that the first breakdown occurs within 8 hours; let say X1 is a random variable representing the interarrival time or interarrival occurrence time of breakdowns. By definition the interarrival times are ~Exponential, f(t) = lamda*exp(-lamda*t), 0 <= t < infinity
Then P{X1 <= 8} = 1 - exp[-(0.5/8)*8] = 1 - exp(-0.5)

b. It will work for at least 4 hours without breakdown.
Here we look for the probability of no breakdown in at least 4 hours, or a breakdown in at most 4 hours.
P{N(4) = 0} = exp[-(0.5/8)*4]
P{X1 > 4} = 1 - P{X1 <= 4} = exp[-(0.5/8)*4]

I hope it helps.
 
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What is the Poisson Distribution?

The Poisson Distribution is a probability distribution that is used to model the number of occurrences of a certain event within a specific time or space, assuming that the events occur independently and at a constant rate.

What are the key characteristics of the Poisson Distribution?

1. The number of events occurring in a given interval is independent of the number of events occurring in any other non-overlapping interval.2. The probability of an event occurring is constant throughout the interval.3. The average number of events occurring in the interval is known.4. The events are considered to be rare.

How is the Poisson Distribution different from other probability distributions?

The Poisson Distribution is different from other probability distributions in that it is used to model rare events, while other distributions may be used for more common events. It also assumes that the events are independent and occur at a constant rate, whereas other distributions may have different assumptions.

What are some real-world applications of the Poisson Distribution?

The Poisson Distribution is often used in fields such as biology, economics, and engineering to model rare events such as the number of mutations in DNA, the number of accidents in a factory, or the number of customer arrivals at a bank.

How do I calculate probabilities using the Poisson Distribution?

To calculate probabilities using the Poisson Distribution, you will need to know the average number of events occurring in the given interval and then use the appropriate formula. Alternatively, you can use statistical software or a Poisson Distribution calculator to calculate probabilities.

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