Search results
Results from the WOW.Com Content Network
After analyzing the data, if the p-value is less than α, that is taken to mean that the observed data is sufficiently inconsistent with the null hypothesis for the null hypothesis to be rejected. However, that does not prove that the null hypothesis is false. The p-value does not, in itself, establish probabilities of hypotheses. Rather, it is ...
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
In more formal probability theory, a random variable is a function X defined from a sample space Ω to a measurable space called the state space. [2] [a] If an element in Ω is mapped to an element in state space by X, then that element in state space is a realization.
For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random ...
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
Here F X is the cumulative distribution function of X, f X is the corresponding probability density function, Q X (p) is the corresponding inverse cumulative distribution function also called the quantile function, [2] and the integrals are of the Riemann–Stieltjes kind.
Conditional probability may be treated as a special case of conditional expectation. Namely, P ( A | X) = E ( Y | X) if Y is the indicator of A. Therefore the conditional probability also depends on the partition α X generated by X rather than on X itself; P ( A | g(X) ) = P (A | X) = P (A | α), α = α X = α g(X).
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.. The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability ...