What does it mean for a random variable to "admit" a distribution? Can someone explain the word "admit" and explain what would happen if it does not admit a distribution?
Not every probability distribution on the real numbers can be described by a probability density function: for example, the probability distribution given by (on the variable $X$)

$$ P(X = a) = \begin{cases} 1/2 &amp; a=0,1 \\ 0 &amp; \text{otherwise} \end{cases} $$

cannot be expressed by a probability density function.

"Admit" is sometimes used to express the notion that you're allowed or able to do something -- in this case, saying that a distribution "admits a density function" means that the distribution the phrase refers to really does have a probability density function.
A random variable, $X$, is a measurable function from a probability space $(\Omega,\mathcal{F},\mathbb{P})$ to $(\mathbb{R},\mathcal{B}_{\mathbb{R}})$. The induced measure on $\mathcal{B}_{\mathbb{R}}$, defined by $$\mu(A)=\mathbb{P}(X^{-1}(A))$$ is the distribution of $X$.