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Suppose that $X$ is a random variable with PDF $f(x)$ with support $(-\infty,\infty)$. Suppose that the expectation of $X$ is $\mathbb{E}(X)=\lambda$.
Is it always true that
$$\int_{-\infty}^{\lambda} f(x)dx = \frac{1}{2}$$ $$\int_\lambda^{\infty}f(x)dx = \frac{1}{2}$$
No.
In your integral equations, $\lambda$ is the median, not the mean. It may be the case that the median and mean are equal (such as a normal distribution), but they do not have to be.
As a counterexample, consider $X\sim exp(1)$.
I'm not surprised that you struggle with the proof, because this does not hold.
As a simple counterexample (with support that is really the whole real line), consider a mixture of two normals with different means and unequal weights. For instance, $0.25\times N(0,0.1)+0.75\times N(1,0.1)$ has a mean of $0.75$, but:
> library(EnvStats)
> pnormMix(q=0.75,mean1=0,sd1=.1,mean2=1,sd2=.1,p.mix=0.25)
[1] 0.7515524
p.mix refers to the first component, with mean1=0 and sd1=0.1. Am I missing something?
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Feb 3, 2021 at 7:03
It is not true since, as others have said, the median does not have to be equal to the mean.
What is true with $\mathbb E[X]=\lambda$, if you use the cumulative distribution function $F(x)$, is
$$\int_{-\infty}^{\lambda} F(x)\,dx = \int_\lambda^{\infty}(1-F(x))\,dx $$ so with the density function $$\int_{x=-\infty}^{\lambda}\int_{y=-\infty}^{x} f(y)\,dy\,dx = \int_{x=\lambda}^{\infty}\int_{y=x}^{\infty} f(y)\,dy\,dx $$
No, but that happens in some cases for instance in the Gaussian.
In fact, you have defined the median: the data point for which half of the population (of the dataset) is higher compared to this value (and therefore the other half being lower to that same value).
You have also pointed to a nice qualitative property of probability distributions:
Consider you have positive numbers drawn from a "heavy tailed" distribution, like wealth in a population of individuals. The more you have inequalities (that is, a high population of poor people and some very wealthy outliers), then the lower this median will be compared to the mean. This defines a shape parameter which is qualitatively important to describe probability distribution functions.
