# Linear Probability Functions Family

A Linear Probability Functions Family is a probability function family that is a linear function family.

**AKA:**Linear Statistical Distribution.**Context:**- …

**Counter-Example(s):****See:**Bernoulli Trial, Unit Interval, Linear Regression Algorithm. Binomial Regression, Simple Linear Regression, Least Squares, Weighted Least Squares.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/linear_probability_model Retrieved:2015-2-16.
- In statistics, a
**linear probability model**is a special case of a binomial regression model. Here the observed variable for each observation takes values which are either 0 or 1. The probability of observing a 0 or 1 in any one case is treated as depending on one or more explanatory variables. For the "linear probability model", this relationship is a particularly simple one, and allows the model to be fitted by simple linear regression.The model assumes that, for a binary outcome (Bernoulli trial), [math]Y[/math], and its associated vector of explanatory variables, [math]X[/math],

^{[1]}:[math] \Pr(Y=1 | X=x) = x'\beta . [/math]For this model, :[math] E[Y|X] = \Pr(Y=1|X) =x'\beta,[/math]

and hence the vector of parameters β can be estimated using least squares. This method of fitting would be inefficient.

^{[1]}This method of fitting can be improved by adopting an iterative scheme based on weighted least squares,^{[1]}in which the model from the previous iteration is used to supply estimates of the conditional variances, [math]Var(Y|X=x)[/math], which would vary between observations. This approach can be related to fitting the model by maximum likelihood.^{[1]}A drawback of this model for the parameter of the Bernoulli distribution is that, unless restrictions are placed on [math] \beta [/math], the estimated coefficients can imply probabilities outside the unit interval [math] [0,1] [/math]. For this reason, models such as the logit model or the probit model are more commonly used.

- In statistics, a

- ↑
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