Search results
Results From The WOW.Com Content Network
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
Also, the characteristic function of the sample mean X of n independent observations has characteristic function φ X (t) = (e −|t|/n) n = e −|t|, using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself.
Suppose that is a random variable sampled from the standard normal distribution, where the mean is and the variance is : (,). Now, consider the random variable Q = Z 2 {\displaystyle Q=Z^{2}} . The distribution of the random variable Q {\displaystyle Q} is an example of a chi-squared distribution: Q ∼ χ 1 2 {\displaystyle \ Q\ \sim \ \chi ...
The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions , i.e. the distributions of each of the individual random variables and the conditional probability distributions , which deal with how the outputs of one random variable are distributed when ...
This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in fact this property characterizes the normal distribution – for no other distribution are the sample mean and sample variance independent. [3]
Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter.
For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance).