In a small metropolitan area annual losses due to storm, fire and
theft are independently distributed random variables. The probability density functions are:

Water
f(x) = 0.48*exp(-0.48x)

Fire
f(x) = 0.14*exp(-0.14x)

Theft
f(x) = 0.75*exp(-0.75x)


Determine the probability that the minimum loss does not exceed 0.5.

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My strategy would be to obtain the joint density function by multiplying each density function together (by independence). Then integrate from 0 to 0.5 for each variable. Thoughts?