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Cars pass through a road junction according to Poisson distribution.An average of 7 cars per minute pass through this junction.

What is the expected number of cars passing through in 3 minutes? My answer is $3 \times 7=21$

What is the probability that exactly the expected number pass through the certain 3 minute period? Do I just use the Poisson distribution formula and substitute mean $=x=21$?

So is my equation below correct?$$\frac{ (exp-21)×21^{21}}{21!}$$

nonuser
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Sook Lim
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2 Answers2

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Yes, you should indeed use the Poisson distribution formula, with $\mu=21$, as the problem states and $x=21$: $$P(x,\mu) = \frac{e^{-\mu }\cdot \mu^{x}}{x!} = \frac{1}{e^{\mu}}\frac{\mu^x}{x!}$$ $$\implies P(x,\mu) = \frac{\mu^x}{e^{\mu}x!}$$ $$P(x, \mu) = e^{-21}\frac{21^{21}}{21!}$$ $$P(x, \mu) = \frac{1}{e^{21}}\cdot \frac{21^{21}}{21!}$$ $$P(x, \mu) = \frac{21^{21}}{e^{21}\cdot 21!}$$

Through calculus, substituting $e$ with $2.71828$, we get:

$$P(x,\mu)\approx 0.08671$$

Mr. Xcoder
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  • But why isn't the mean 21 as well?If we use 7,that's the mean for one minute but shouldn't we use the mean for 3 mins which is 21 cars?Or is mean not equal to average? – Sook Lim May 16 '17 at 16:19
  • @SookLim corrected. The problem clearly states that $x=\mu$, sorry, it was my fault – Mr. Xcoder May 16 '17 at 16:25
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$e$: A constant equal to approximately $2.71828$.

$\mu$: The mean number of successes that occur in a specified region.

$x$: The actual number of successes that occur in a specified region.

$P(x; \mu)$: The Poisson probability that exactly $x$ successes occur in a Poisson experiment, when the mean number of successes is $\mu$.

$ e = e$

$\mu = 21$

$x = 21$

$$P(x;\mu) = \frac{e^{-\mu}\times \mu^x}{x!}=\frac{21^{21} \times e^{21}}{21!}$$

So your solution is $$\frac{21^{21} \times e^{-21}}{21!}$$

Xetrov
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  • Be careful to $\mu$. The mean is $21$ and the problem is asking what is the probability that the conditions are met when $\mu = x$, so $\mu = 21$. "What is the probability that exactly the expected number pass through the certain 3 minute period" – Mr. Xcoder May 16 '17 at 16:27