Discrete Random Variable Probabilities?

Question

I have the following discrete random variable $\mathbb{X}$ over $\Omega = \{1,2,3,4,5\}$ with the pmf:

$f_{\mathbb{X}}(1) = P(\mathbb{X} = 1) = 0.25$

$f_{\mathbb{X}}(2) = P(\mathbb{X} = 2) = 0.4$

$f_{\mathbb{X}}(3) = P(\mathbb{X} = 3) = 0.1$

$f_{\mathbb{X}}(4) = P(\mathbb{X} = 4) = 0.15$

$f_{\mathbb{X}}(5) = P(\mathbb{X} = 5) = 0.1$

and I want to find $P(\mathbb{X} = \mu)$

I can easily find $\mu$ as follows:

$$\mu = \sum_{i = 1}^5 xf_{\mathbb{X}}(x) = 1(0.25)+2(0.4)+3(0.1)+4(0.15)+5(0.1) = 2.45$$

but, how do you find $P(\mathbb{X} = 2.45)$?

Do I simply round down since $\mathbb{X}$ is discrete? Do I interpolate? Any clarification would be great!

Answer 1

You need to assume a functional form for the distribution:

A simple quadratic fit gives: $f(X) = 0.34 - 0.0335714 x - 0.00357143 x^2$ and then $f(2.45) = 0.236313$.