f(x)=9xe^-(3x) where x is greater than or equal to 0, AND f(x)=0 for all other v

Question

f(x)=9xe^-(3x) where x is greater than or equal to 0, AND f(x)=0 for all other values of x. Show that it is a probability density function.

Please show all work. I think I may be skipping a step or doing something wrong in my integration by parts.

Explanation / Answer

Assuming you mean 9xe^(-3x) instead of 9xe^-3x, 8 ? 9xe^(-3x) dx = 0 lim .....a a-> 8..s 9xe^(-3x) dx = ..........0 u = -3x, du = -3dx dv = -3e^(-3x) dx, v = e^(-3x) ................................a lim -3xe^(-3x) - e^(-3x) | = a-> 8........................0 lim -3ae^(-3a) - e^(-3a) + 1 = 0 - 0 + 1 = 1 a-> 8 Since the integral equals 1, the function is a probability density function.

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