What's the probability that exactly $12$ buses will arrive within $12$ hours
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Let's suppose there are two buses number $86$ and $98$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the $15$th bus will arrive?, (b)What's the probability that exactly $12$ buses will arrive within $3$ hours?
Poisson distribution $P(N(t)=j)=frac{(lambda t)^j}{j!}e^{-lambda t}$. We have that $j =3$ or $j =5$. Do I just substitute $j =3$ and $lambda t=15$ and we immediately have (a)? I'm aware that's a really easy exercise but I somehow don't really know how to approach this one. I'm also not sure how to approach subpoint (b). I'll be thankful for any tips and help.
probability-theory stochastic-processes
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
$endgroup$
add a comment |
$begingroup$
Let's suppose there are two buses number $86$ and $98$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the $15$th bus will arrive?, (b)What's the probability that exactly $12$ buses will arrive within $3$ hours?
Poisson distribution $P(N(t)=j)=frac{(lambda t)^j}{j!}e^{-lambda t}$. We have that $j =3$ or $j =5$. Do I just substitute $j =3$ and $lambda t=15$ and we immediately have (a)? I'm aware that's a really easy exercise but I somehow don't really know how to approach this one. I'm also not sure how to approach subpoint (b). I'll be thankful for any tips and help.
probability-theory stochastic-processes
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
$endgroup$
4
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
1
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10
add a comment |
$begingroup$
Let's suppose there are two buses number $86$ and $98$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the $15$th bus will arrive?, (b)What's the probability that exactly $12$ buses will arrive within $3$ hours?
Poisson distribution $P(N(t)=j)=frac{(lambda t)^j}{j!}e^{-lambda t}$. We have that $j =3$ or $j =5$. Do I just substitute $j =3$ and $lambda t=15$ and we immediately have (a)? I'm aware that's a really easy exercise but I somehow don't really know how to approach this one. I'm also not sure how to approach subpoint (b). I'll be thankful for any tips and help.
probability-theory stochastic-processes
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
$endgroup$
Let's suppose there are two buses number $86$ and $98$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the $15$th bus will arrive?, (b)What's the probability that exactly $12$ buses will arrive within $3$ hours?
Poisson distribution $P(N(t)=j)=frac{(lambda t)^j}{j!}e^{-lambda t}$. We have that $j =3$ or $j =5$. Do I just substitute $j =3$ and $lambda t=15$ and we immediately have (a)? I'm aware that's a really easy exercise but I somehow don't really know how to approach this one. I'm also not sure how to approach subpoint (b). I'll be thankful for any tips and help.
probability-theory stochastic-processes
probability-theory stochastic-processes
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
edited Jan 11 at 16:46
MacAbra
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
asked Dec 23 '18 at 17:36
MacAbraMacAbra
269210
269210
4
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
1
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10
add a comment |
4
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
1
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10
4
4
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
1
1
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10
add a comment |
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4
$begingroup$
Hint: Although we have two buses, the question considers the arrival times/arrival of buses in general. If bus A arrives on average $3$ times per hour and bus B arrives on average $5$ times per hour, then a bus comes to the bus stop on average $8$ times per hour.
$endgroup$
– J. Pistachio
Dec 23 '18 at 18:52
1
$begingroup$
@J.Pistachio I've tried to use a central limit theorem for (b), but I have to find the probability that exactly $12$ buses will arrive ( $P(S_n=12)$ ), so I'm not sure if I can use this theorem (I need to use a standard normal table then).
$endgroup$
– MacAbra
Jan 5 at 17:10