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I know that for a simple linear regression the predictions are distributed like:

$$y_i\, |\, x_i\, \sim\, \mathcal{N}\big(\widehat{\beta}_0+\widehat{\beta}_1\, x_i,\ \sigma^2\big)$$ $$\text{where: } \sigma^2 = \frac{1}{n} \sum_{i=1}^n \big(y_i - (\widehat{\beta}_0+\widehat{\beta}_1\, x_i)\big)^2$$

However I want to fit an quasi-linear regression using R's glm function with the variance being proportional to the mean, as follows:

my_fit <- glm(my_ys ~ my_xs, family=quasi(link="identity", variance="mu"))

This returns a slope and an intercept as before, but it also returns a dispersion parameter, which I'll call $\theta$. Then am I correct that predictions from this model will be distributed like:

$$y_i\, |\, x_i\, \sim\, \mathcal{N}\big(\widehat{\beta}_0+\widehat{\beta}_1\, x_i,\ \theta \cdot (\widehat{\beta}_0+\widehat{\beta}_1\, x_i) \cdot \sigma^2\big)$$ $$\text{where: } \sigma^2 = \frac{1}{n} \sum_{i=1}^n \big(y_i - (\widehat{\beta}_0+\widehat{\beta}_1\, x_i)\big)^2$$

I've never used fitted a "quasi" regression before, so I want to be sure that my understanding is correct before I go making recommendations based upon my analysis.

EDIT

Based on the answer here, it's clear that the dispersion parameter just takes the place of $\sigma^2$, so I believe that the form of the predictions is actually:

$$y_i\, |\, x_i\, \sim\, \mathcal{N}\big(\widehat{\beta}_0+\widehat{\beta}_1\, x_i,\ \theta \cdot (\widehat{\beta}_0+\widehat{\beta}_1\, x_i)\big)$$

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