## Question

###### There are two types of claims that are made to an insurance company. Let $N_{i}(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $left{N_{1}(t), t geqslant 0ight}$ and $left{N_{2}(t), t geqslant 0ight}$ are independent Poisson processes with rates $lambda_{1}=10$ and $lambda_{2}=1 .$ The amounts of successive type 1 claims are independent exponential random variables with mean $$ 1000$ whereas the amounts from type 2 claims are independent exponential random variables

There are two types of claims that are made to an insurance company. Let $N_{i}(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $left{N_{1}(t), t geqslant 0 ight}$ and $left{N_{2}(t), t geqslant 0 ight}$ are independent Poisson processes with rates $lambda_{1}=10$ and $lambda_{2}=1 .$ The amounts of successive type 1 claims are independent exponential random variables with mean $$ 1000$ whereas the amounts from type 2 claims are independent exponential random variables with mean $$ 5000 .$ A claim for $$ 4000$ has just been received; what is the probability it is a type 1 claim?

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