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I have a problem understanding the first part of this question :

In a certain town, at time $t = 0$ there are no bears. Brown bears and grizzly bears arrive as independent Poisson processes with respective rates $\beta$ and γ.

(a) What is the probability that a brown bear is the first to arrive?

I know that I have to calculate $P( T < U)$ where $T$ and $U$ are respectively random variables describing the time arrival of brown and grizzly bears, but this seems weird for me.

Alejandro Tolcachier
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yassinemab
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  • Why does it seem weird to you? – Matti P. Jan 15 '20 at 10:25
  • it seems weird since we usually work out the probability that a random variable (upper case) is smaller than a realization (lower case) such as P( T< t), Then can apply the formula. But in this case, I don't understand how to calculate this probability. – yassinemab Jan 15 '20 at 10:29
  • Grizzly bears are brown bears. Just saying. Black bears are different. – lulu Jan 15 '20 at 10:45
  • @lulu : What's the Difference Between Grizzly Bears. ... The difference is regional: bears found inland are referred to as grizzlies, while those on the coasts are known as brown bears. Grizzlies are actually a subspecies of brown bear, Ursus arctos horribilis, found in dense forests, alpine meadows and mountain valleys. – yassinemab Jan 15 '20 at 10:51
  • @yassinemab Exactly. All grizzly bears are brown bears. Not all brown bears are grizzly bears. Probably not the point of the exercise, but still. – lulu Jan 15 '20 at 10:54
  • @lulu : yes you'r right. – yassinemab Jan 15 '20 at 10:55

1 Answers1

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$T$ and $U$ are independent exponential random variables with parameters $\beta$ and $\gamma$ so $P(T<U)=\int_0^{\infty} \int_0^{u}\beta e^{-\beta t} dt \gamma e^{-\gamma u} du$. I will let you evaluate this integral.

Kavi Rama Murthy
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  • Thanks a lot, I didn't know this formula, can you please tell me how to find it? – yassinemab Jan 15 '20 at 10:32
  • @yassinemab ,they are independent; so, imagine the 2D plane of the variable T and U. Over which region does it seem reasonable to integrate and find the probability? Given that T<U? – Someone Jan 15 '20 at 14:12