Question

Let T1 be the time between a car accident and reporting a claim to the insurance company. Let T2 be the time between the report of the claim and payment of the claim. The joint density function of T1 and T2, f(t1, t2), is constant over the region 0 < t1 < 6, 0 < t2 < 6, t1 t2 < 10, and zero otherwise. Determine E[T1 T2], the expected time between a car accident and payment of the claim.

Answer 1

5.7255

Step-by-step explanation:

From the given information:

`T_1 \to \text{time between car accident \& reporting claim} \\ \\ T_2 \to \text{time between reporting claim and payment of claim}`

The joint density function of
`T_1` and
`T_2` is:

`f(t_1,t_2) = \left \{ {{c \ \ \ 0<t_1<6, \ \ \ 0<t_2<6, \ \ \ t_1+t_2<10} \atop {0} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise} \right.`

Area(A):
`= 6* 6 - (1)/(2)*2*2`

= 34

The limits are:

`\text{limits of } \ t_1 \ from \ 0 \ is \ 10 \to t_2 \\ \\ \text{limits of } \ t_2 \ from \ 0 \ is \ 4 \to 6`

Also;

`\text{limits of } \ t_1 \ is \ 0 \to 6 \\ \\ \text{limits of } \ t_2 \ is \ 0 \to 4`

∴

`\iint f(t_1,t_2) dt_1dt_2 =1 \\ \\ c \iint 1dt_1dt_2 = 1 \\ \\ cA = 1 \\ \\ \implies c = (1)/(34)`

To find;

`E(T_1+T_2) = \iint (t_1+t_2)c \ \ dt_1dt_2 \\ \\ \implies (1)/(34) \Big[\int \limits^4_0 \int \limits^6_0(t_1+t_2) dt_1 \ dt_2 + \int \limits^6_4 \int \limits^(10-t_2)_0(t_1+t_2) dt_1 dt_2 \Big] \\ \\ \implies (1)/(34) (120 + (224)/(3)) \\ \\ = \mathbf{5.7255}`