Answer:
5.7255
Explanation:
From the given information:
[tex]T_1 \to \text{time between car accident \& reporting claim} \\ \\ T_2 \to \text{time between reporting claim and payment of claim}[/tex]
The joint density function of [tex]T_1[/tex] and [tex]T_2[/tex] is:
[tex]f(t_1,t_2) = \left \{ {{c \ \ \ 0<t_1<6, \ \ \ 0<t_2<6, \ \ \ t_1+t_2<10} \atop {0} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise} \right.[/tex]
Area(A): [tex]= 6\times 6 - \dfrac{1}{2}*2*2[/tex]
= 34
The limits are:
[tex]\text{limits of } \ t_1 \ from \ 0 \ is \ 10 \to t_2 \\ \\ \text{limits of } \ t_2 \ from \ 0 \ is \ 4 \to 6[/tex]
Also;
[tex]\text{limits of } \ t_1 \ is \ 0 \to 6 \\ \\ \text{limits of } \ t_2 \ is \ 0 \to 4[/tex]
∴
[tex]\iint f(t_1,t_2) dt_1dt_2 =1 \\ \\ c \iint 1dt_1dt_2 = 1 \\ \\ cA = 1 \\ \\ \implies c = \dfrac{1}{34}[/tex]
To find;
[tex]E(T_1+T_2) = \iint (t_1+t_2)c \ \ dt_1dt_2 \\ \\ \implies \dfrac{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 \dfrac{1}{34} (120 + \dfrac{224}{3}) \\ \\ = \mathbf{5.7255}[/tex]
