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In bayesian inference we know that in order to make predictions for future observations we must to calculate posterior predictive distribution.

$$f(y|x)=\int f(y|\theta) p(\theta|x)d\theta $$

where $p(\theta|x)$ is the posterior distribution of $\theta$ and $f(y|\theta)$ is the distribution of our data. On other hand if we approach the problem of prediction with classical statistical inference then we will just have to take the normal distribution with mean the $mean(Data)$ and variance $\sqrt{variance(Data)/n}$

Is this the only way in classical statistical inference to make predictions with the use of distribution ?? (except C.I or point estimations)

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No. For instance you can assume that your data comes from a parameterised family of distributions and try to learn the parameters, e. g. with maximum likelihood. Or you can use non parametric techniques like kernel density estimation. You can also frame your problem as a regression and assume some parameterised functional dependence (e.g. linearity) plus noise, typically Gaussian with zero mean, and try to learn the parameters of the dependence with any of a number of algorithms. There are many different options. You can read more about all this in any statistics book, like Casella's or Wassermann's.

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