Gamma distribution shape

Question

I have a gamma distribution with the following pdf: $$ f(x) = \frac{1}{4} xe^{-0.5x}, x > 0$$

I am trying to determine the shape of the graph without plotting it. I am given a hint ot consider the mean and standard deviation.

I have calculated the mean is $\mu = 4$ and standard deviation is $\sigma = 2\sqrt{2}$, so the standard deviation is large relative to its mean, but I am unable to tell what this mean exactly. Is it positively/negatively skewed based of this?

Answer 1

The large-$x$ limit is dominated by the exponential term. and extends out infinitely far. (The distribution is bounded by $x = 0$ at the left.) As such, the distribution must be skewed positively. So you know (without graphing):

• $f(x=0)=0$
• ${\rm Mean}[f(x)] = 4$
• There is a single maximum of $f(x)$ (and no inflection points)
• The distribution tails off to infinity
• Hence the distribution is skewed positively
• The standard deviation is $\sigma = 2 \sqrt{2}$
• $\int\limits_{x=0}^\infty f(x)\ dx = 1$ (of course)

Isn't this enough? Just sketch $f(x)$ from this knowledge and you'll be very close.