# What is $\mu_i$ in a GLM / link function

Is it so that:

• $y_i$ is not a discrete value, but a range with probability density function
• Which means for the same predictor(s) value $y_i$ could have different results
• In linear regression this distribution can only be normal
• In GLM, this distribution can be any distribution from the exponential family
• distribution of a single $y_i$ has nothing to do with distribution of all $y(s)$
• $\mu_i$ is expected value of $y_i$
• In practical use, $\mu_i$ is the predicted value $y_i$, specially if dataset has only one y for given predictor(s)

Are above correct? Where am I wrong?

Based on the above I've tried simulating glm with lm in R, and it kinda works:

library(boot)
# calling logit(ravensData$ravenWinNum) results in # [1] Inf Inf Inf Inf Inf -Inf Inf Inf Inf Inf -Inf Inf Inf Inf Inf -Inf # [17] -Inf -Inf Inf -Inf # that's way too much, as inv.logit goes to 1 at 20 # so we'll write our own dummy "logit" routine # this will give us 5 when winNum=1 and -5 when it's zero win <- ravensData$ravenWinNum*10-5
fit <- lm(win~ravensData$ravenScore) # and get probability of win using inv.logit fitwin <- inv.logit(fit$fitted.values)