I have the following exercise:
The occurence of a claim follows an exponential distribution with mean 4000. The insurance imposes an upper limit of 6000 for coverage in the policy. If a claim occurs for sure, the payment of the policy has the following cdf:
F(x) =
0 if x <0
1-e^(-x/4000) if 0 ≤ x < 6000
1 if x≥6000.
a) Check at which points F is continuous. Draw a graph of the function.
b)Bonus question(useful for next semester): Compute the expectation of the loss.
I want to be sure if my way to check for continuity is right. For me a function is continuous at point c if f(c) = lim x->c f(x). So in that case:
- f(0)= 0, lim x->0 f(x)= lim x-> 1-e^(-x/4000)=0 (e to the power 0 = 1).
--> So in that case the function is continuous at x=0
- f(6000)=1, lim x->6000 f(x)= lim x->6000 1-e^(-6000/4000)= 0.776
--> So in that case the function is discontinuous at x=6000
I'm scared I missed something by not checking for each situation the limit from the right and the limit from the left. Am I right to do so ?