What is the distribution of changes in adjusted R squared when adding a random normal variable? Suppose we have a model with $n$ data points and $p$ predictors.  The model has an $R^2_{\text{adjusted}}=q$.
What sort of distribution does the change in $R^2_{\text{adjusted}}$ have when a random Gaussian predictor is added to the model?
I am assuming that the expected value is zero, since $R^2_{\text{adjusted}}$ is supposed to adjust for predictors that have no explanatory power.  But I can't find the answer anywhere, and I'm having trouble working it out myself.
...
 A: Here's a quick  simulation in R that gives you a feel for the distribution of $R^2_{adj}$ when adding a single random normal variable.
set.seed(1234)
adj_r_simulation <- function(n=100) {
    y <- rnorm(n)
    x <- rnorm(n)
    fit <- lm(y~x)
    summary(fit)$adj.r.squared
}

# Simulate adjusted r-square values
simulations <- 1000
n <- 100
x <- sapply(seq(simulations), function(X) adj_r_simulation(n=n))

# Display results
mean(x)
sd(x)
plot(density(x), 
     main=paste0("n=",n, "; simulations=", simulations),
     xlab='Adjusted r-squared')
abline(v=mean(x))

The simulation above simulates two independent normal variables, x and y of sample size n=100. It then fits a linear model predicting y from x and returns $R^2_{adj}$. This process is repeated 1000 times and the values are saved. The sample mean of simulated $R^2_{adj}$ values is close to zero:
> mean(x)
[1] 0.0002181863
> sd(x)
[1] 0.01425647

Below you can see the density plot of the simulated values of $R^2_{adj}$. This illustrates whuber's point about the distribution being highly skewed.

