Expectation of log skew normal distribution

Question

What is the expected value and expected variance of a log skew normal distribution?

In case I have the terminology wrong, I'm referring to data that is lognormal with some skew mild skew when it's log transformed so it would have params y ~ logskewnormal(loc, scale, alpha).

edit: Got a great answer from JimB, code below to reproduce the expected value of the mean!

import numpy as np
import scipy.stats as stats
from scipy.special import erf, erfc

a, mu, sig = 1.1, 3.9, 1.25
skewnorm_dist = stats.skewnorm(a=a, loc=mu, scale=sig)

exp_mu_hat = []
for _ in range(10000):
    X = skewnorm_dist.rvs(1000)
    Y = np.exp(X)
    exp_mu_hat.append( Y.mean() )
    
print( np.mean(exp_mu_hat) )

term1 = np.exp(mu + 0.5*(sig**2))
term2 = erf((a*sig) / (np.sqrt(2) * np.sqrt(a**2 + 1))) + 1
exp_mu = term1*term2
print(exp_mu)

Answer 1

I wonder if you mean the following:

If $X\sim N(\mu,\sigma^2)$, then $Y=e^X$ has a log normal distribution.

So if $X\sim \text{SkewNormal}(\mu,\sigma,\alpha)$, then are you really interested in $Y=e^X$?

If so and the pdf of $X$ is defined to be

$$\frac{\sqrt{\frac{2}{\pi }} e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}} \Phi \left(\frac{\alpha (\log (x)-\mu )}{\sigma }\right)}{\sigma x}$$

then the mean and variance are

$$\mu_Y=2 e^{\mu +\frac{\sigma ^2}{2}} \left(1-\Phi \left(-\frac{\alpha \sigma }{\sqrt{\alpha ^2+1}}\right)\right)$$

$$\sigma^2_Y=2 e^{2 \mu +\sigma ^2} \left(e^{\sigma ^2} \left(1-\Phi \left(-\frac{2 \alpha \sigma }{\sqrt{\alpha ^2+1}}\right)\right)-2 \left(1-\Phi \left(-\frac{\alpha \sigma }{\sqrt{\alpha ^2+1}}\right)\right)^2\right)$$

where $\Phi(.)$ is the cdf of the standard normal distribution.