Question

A random variable X is gamma distributed with α = 3, β = 2. Find: P(l ≤ X ≤ 2)

Answer 1

If the random variable has a distribution gamma, then the distribution function has the form

$F\left( {x,\alpha ,\beta } \right) = 1 - \sum\limits_{i = 0}^\alpha {{e^{ - \frac{x}{\beta }}}} {\left( {\frac{x}{\beta }} \right)^i}\frac{1}{{i!}} = 1 - \sum\limits_{i = 0}^3 {{e^{ - \frac{x}{2}}}} {\left( {\frac{x}{2}} \right)^i}\frac{1}{{i!}},\,\,x > 0$

Then

$P(1 \le x \le 2) = F(2) - F(1) = 1 - \sum\limits_{i = 0}^3 {{e^{ - \frac{2}{2}}}} {\left( {\frac{2}{2}} \right)^i}\frac{1}{{i!}} - \left( {1 - \sum\limits_{i = 0}^3 {{e^{ - \frac{1}{2}}}} {{\left( {\frac{1}{2}} \right)}^i}\frac{1}{{i!}}} \right) = {e^{ - \frac{1}{2}}}\left( {1 + \frac{1}{2} + \frac{1}{8} + \frac{1}{{48}}} \right) - {e^{ - 1}}\left( {1 + 1 + \frac{1}{2} + \frac{1}{6}} \right) \approx 0.017$

Answer: $P(1 \le x \le 2) \approx 0.017$