X is an exponential random variable

Question

for this I did $$ln(1) - ln(P(x^2 + 10x - 24 \geq 0 ) = ln(P(-12 < x < 2) = ln(1 - (1 - e^{-18}) + (1 - e^{108})) $$

When I put this into the calcuator I get error as the answer. What did I do wrong?

Answer 1

You should get $-ln P(X\geq 2)$ becasue $x^{2}+10x \geq 24$ iff $|x+5| \geq 7$ iff $x \geq 2$ or $x \leq -12$ and $x \geq 0$ in our case. Can you continue from here?

Note that exponential random variables are positive. You are missing this point in your calculation.