Is the integral of the probability density of finding a paricle symmetric about the mean? Is the probability density integral of finding a particle symmetric about the expectation value? Also is the probability integral symmetric with respect to standard deviations from the mean? Like if I calculate the probability of finding a particle to the right of $x$ plus one standard deviation, is it the same as the probability of finding it to the left of $x$ minus one standard deviation? $x$ here is the expectation value.
 A: 
Is the probability density integral of finding a particle symmetric about the expectation value?

No, not necessarily. Some probability distributions are symmetric and others are not (these are called skewed distributions).
One should be particularly conscious about the difference between the mean, the median and the mode of a distribution:

*

*mean is the expectation value, i.e., $$\mu=\int_{-\infty}^{+\infty} x\rho(x)dx $$

*median is the value that the probability to be above the median is the same as the probability to be below the median, i.e., $$\int_{-\infty}^m\rho(x) dx=\int_m^{+\infty}\rho(x) dx$$

*mode is the highest point of the distribution, i.e., the point of maximum probability density (a distribution may have more than one mode - multimodal distribution).


Also is the probability integral symmetric with respect to standard deviations from the mean? Like if I calculate the probability of finding a particle to the right of x+ one standard deviation, is it the same as the probability of finding it to the left of x-one standard deviation?

Standard deviation is a parameter of normal distribution, i.e., of the distribution with probability density
$$\rho(x|\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$
By the Central limit theorem, many repeated measurements of continuous quantities can be described by a normal distribution, and this is the reason why the standard deviation, $\sigma^2$, is frequently used to describe the spread of a distribution. In QM we often deal with distributions that are not normal. (Remark: the ground state of a harmonic oscillator is true normal distribution).
Kurtosis and skewness are quantities often used in statistics to described deviations from normality and asymmetry of distributions.
