1

Seeking the expected value of a continuous random variable we calculate the integral $\int_{-\infty}^\infty xf_X(x)\ dx$.
Does the integrand $xf_X(x)$, i.e. the product of $x$ and the corresponding density, have any (interesting) interpretation?

(An analogous question could be posed for the case of a discrete random variable.)

Richard Hardy
  • 54,375
  • 10
  • 95
  • 219
  • One interpretation of the above integral is to be the limiting value of an empirical average. – Xi'an Jan 14 '21 at 08:00
  • @Xi'an, thank you. Any interpretations of the integrand, though? – Richard Hardy Jan 14 '21 at 08:08
  • In the discrete case, $kf_X(k)$ is the limiting value of $k\hat{f_n}(k)$ when $\hat{f_n}(k)$ is the frequency of $k$'s in the $n$-sample. – Xi'an Jan 14 '21 at 08:30
  • 1
    You could make plots of *visual means* as analogues of the plot in https://stats.stackexchange.com/questions/84158/how-is-the-kurtosis-of-a-distribution-related-to-the-geometry-of-the-density-fun/362745#362745 , showing where the contributions to the mean comes from. If $f$ is an income distribution, say, this will show where the purchasing power is. Also search for *partial moments*. – kjetil b halvorsen Jan 14 '21 at 10:17

1 Answers1

2

You could see it as the amount of 'leverage' at the point $x$.

In physics, this would be torque which is $weight \times distance$ or $force \times distance$.

lever

Sextus Empiricus
  • 43,080
  • 1
  • 72
  • 161