Gamma Distribution Engineering system fails

Question

Let $Y$ denote the time until an engineering system fails. An engineer models Y as a gamma random variable with $α= 2$ and $β= 2.$ Find $P(Y > 2).$

Using this definition gamma distribution

$$ g(x;α,β) = \left\{ \begin{array}{ll} \frac{1}{β^{α}\Gamma(α) } \cdot x^{α-1}e^{-x/β} & \text{for x > 0}\\ 0 &\text{ elsewhere} \\ \end{array} \ \right. $$

Where $α>0$ and $β>0$

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My attempt Density Function

$$ g(x;2,2) = \left\{ \begin{array}{ll} \frac{1}{{8\Gamma }} \cdot xe^{-x/2} & \text{for y > 0}\\ 0 &\text{ elsewhere} \\ \end{array} \ \right. $$

$P(Y>2) =\int^{2}_{0} \frac{1}{8}xe^{-x/2}dy$

Would this be the correct way of setting the problem up?

Answer 1

Two errors: You left a mysterious $\Gamma$ in the denominator of your distribution; that factor should be
$$ \frac{1}{2^{2\Gamma(2)}} = \frac{1}{4} $$ (you did notice that $\Gamma(2) = 1! - 1$, not $2$, right?)

And your integral for $P(Y>2)$ should go from $2$ to $\infty$.