An insurance company provides buildings and contents insurance to its policyholders.We use X to denote the random variable for annual claims under buildings insurance and Y to denote annual claims under contents insurance. Working in units of $10M, X and Y are known to have a joint density function of the form,

where a is an unknown constant.

a. Determine the value of a.

b. Calculate the probability that the total claims exceeds $60M.

As we will see in Chapter 13, a joint density function p(x, y) is such that the probability that two continuous random variables X and Y take values X ∈ [x₁, x₂] and Y ∈ [y₁, y₂] is given by

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p(x, y) = 0 -y+5) for 0 ≤ x ≤ 5, 0≤ y ≤7 otherwise