In a certain city, the daily consumption of electric power in millions of kilowatt-hours can be treated as a random variable having a gamma distribution with α=3 and β=b. If the power plant of this city has a daily capacity of 12 million kilowatt-hours, what is the probability that this power supply will be inadequate on any given day?
The pdf I gave you is if we're working on . The pdf you gave is if we're working on
You can see that and
So here, it would depend on how you have been taught
As for your mistake of b=2, it doesn't change anything to the result, since b was a constant, and that you can substitute it by any positive value you want
And for the double integration by parts, it's because we have in the integrand. We can't calculate an antiderivative of it. So make an integration by parts : the power x^2 will be transformed into x and the exponential remains. But we still cannot find an antiderivative. So integrate again by parts : the power x will be transformed to a constant and the exponential remains. But then you can compute an antiderivative, because the integrand will be an exponential.
Just do it and you'll see (each time, take the u part as x^2 or x and the dv part as the exponential)