Is there a phrase for to describe $xf(x)$ where $f(\cdot)$ is the probability mass function?

Question

We know that $\mathbb{E}[X] = \sum_x xf_{X}(x)$ where $f_X(\cdot)$ is the probability mass function.

Is there any phrase that we use to describe $\sum_{x\le a} xf_X(x)$, or is there any phrase that we can use to denote $xf_X(x)$?

Answer 1

The magnitude

$$\sum_{x\le a} xf(x) \equiv E(X; X\leq a)$$

is called in some circles a "restricted" expected value (restricted in the interval $(\infty, a]$), in order to keep the familiar "expected value" terminology while making the distinction from the "truncated" expected value,

$$\sum_{x\le a} x\frac{f(x)}{F_X(a)} = E(X \mid X\leq a),$$

where $F_X(a)$ is the distribution function of $X$ evaluated at $a$.

Note the use of the symbol "$;$" in the first case, and the symbol "$\mid $" in the second that has the usual "conditional on" meaning.

The relation is

$$E(X; X\leq a) = E(X \mid X\leq a)\cdot {\rm Pr}(X\leq a).$$

And so the general expression for the decomsposition of the expected value

$$E(X) = E(X \mid X\leq a)\cdot {\rm Pr}(X\leq a) + E(X \mid X> a)\cdot {\rm Pr}(X> a)$$

$$= E(X; X\leq a) + E(X; X> a).$$

Answer 2

The quantity $\sum_{x\leqslant a} x^q p(x)$ if the random variable $X$ is discrete with mass function $p(x)$, or $\int_{-\infty}^a x^q f(x)\text{d}x$ if $X$ is continuous with density $f(x)$, is sometimes referred to as the 'partial' $q$th moment of $X$. If $q=1$, then you could say this is a partial (lower) first moment of $X$. See for example here.