**Analizzare un campione**

First, we create some random variables. We set a ** seed** so that in each run we get identical results to look at. As an example we take a sample from the Student

**distribution:**

`t`

Here, we set the required shape parameter of the ** t** distribution, which in statistics corresponds to the degrees of freedom, to 10. Using

**means that our sample consists of 1000 independently drawn (pseudo) random numbers. Since we did not specify the keyword arguments**

`size=1000`

**and**

`loc`

**, those are set to their default values zero and one.**

`scale`

**statistiche descrittive**

** x** is a numpy array, and we have direct access to all array methods, e.g.

How do the some sample properties compare to their theoretical counterparts?

Note: stats.describe uses the unbiased estimator for the variance, while np.var is the biased estimator.

For our sample the sample statistics differ a by a small amount from their theoretical counterparts.

**test** `t`

**e** `KS`

We can use the t-test to test whether the mean of our sample differs in a statistically significant way from the theoretical expectation.

The ** pvalue** is 0.7, this means that with an alpha error of, for example, 10%, we cannot reject the hypothesis that the sample mean is equal to zero, the expectation of the standard t-distribution.

As an exercise, we can calculate our ** t**-test also directly without using the provided function, which should give us the same answer, and so it does:

The Kolmogorov-Smirnov test can be used to test the hypothesis that the sample comes from the standard t-distribution

Again the ** pvalue** is high enough that we cannot reject the hypothesis that the random sample really is distributed according to the

**-distribution. In real applications, we don’t know what the underlying distribution is. If we perform the Kolmogorov-Smirnov test of our sample against the standard normal distribution, then we also cannot reject the hypothesis that our sample was generated by the normal distribution given that in this example the**

`t`

**is almost 40%.**

`pvalue`

However, the standard normal distribution has a variance of 1, while our sample has a variance of 1.29. If we standardize our sample and test it against the normal distribution, then the ** p**-value is again large enough that we cannot reject the hypothesis that the sample came form the normal distribution.

Note: The Kolmogorov-Smirnov test assumes that we test against a distribution with given parameters, since in the last case we estimated mean and variance, this assumption is violated, and the distribution of the test statistic on which the ** p**-value is based, is not correct.

OK, confesso: sono niubbassay 😊

**code della distribuzione**

Finally, we can check the upper tail of the distribution. We can use the percent point function ** ppf**, which is the inverse of the

**function, to obtain the critical values, or, more directly, we can use the inverse of the survival function**

`cdf`

In all three cases, our sample has more weight in the top tail than the underlying distribution. We can briefly check a larger sample to see if we get a closer match. In this case the empirical frequency is quite close to the theoretical probability, but if we repeat this several times the fluctuations are still pretty large.

We can also compare it with the tail of the normal distribution, which has less weight in the tails:

The ** chisquare** test can be used to test, whether for a finite number of bins, the observed frequencies differ significantly from the probabilities of the hypothesized distribution.

We see that the standard normal distribution is clearly rejected while the standard ** t**-distribution cannot be rejected. Since the variance of our sample differs from both standard distribution, we can again redo the test taking the estimate for scale and location into account.

The ** fit** method of the distributions can be used to estimate the parameters of the distribution, and the test is repeated using probabilities of the estimated distribution.

Taking account of the estimated parameters, we can still reject the hypothesis that our sample came from a normal distribution (at the 5% level), but again, with a ** p**-value of 0.95, we cannot reject the

**distribution.**

`t`

**test speciali per distribuzioni normali**

Since the normal distribution is the most common distribution in statistics, there are several additional functions available to test whether a sample could have been drawn from a normal distribution

First we can test if ** skew** and

**of our sample differ significantly from those of a normal distribution:**

`kurtosis`

These two tests are combined in the normality test

In all three tests the ** p**-values are very low and we can reject the hypothesis that the our sample has

**and**

`skew`

**of the normal distribution.**

`kurtosis`

Since ** skew** and

**of our sample are based on central moments, we get exactly the same results if we test the standardized sample:**

`kurtosis`

Because normality is rejected so strongly, we can check whether the normaltest gives reasonable results for other cases:

When testing for normality of a small sample of ** t**-distributed observations and a large sample of normal distributed observation, then in neither case can we reject the null hypothesis that the sample comes from a normal distribution. In the first case this is because the test is not powerful enough to distinguish a

**and a normally distributed random variable in a small sample.**

`t`

Già detto che sono niubbassay 😊 per queste cose qui?