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Since the probabilities must satisfy p 1 + ⋅⋅⋅ + p k = 1, it is natural to interpret E[X] as a weighted average of the x i values, with weights given by their probabilities p i. In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k ), the weighted average is given by the standard average .
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).
The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions. This can create a subtle difference, but in this example yields the same probability of 0.0437.
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 ...
Sometimes the probability of "the value of for the parameter value " is written as P(X = x | θ) or P(X = x; θ). The likelihood is the probability that a particular outcome is observed when the true value of the parameter is , equivalent to the probability mass on ; it is not a probability density over the parameter .
As there is zero X n+1 or X −1 in (1 + X) n, one might extend the definition beyond the above boundaries to include () = when either k > n or k < 0. This recursive formula then allows the construction of Pascal's triangle , surrounded by white spaces where the zeros, or the trivial coefficients, would be.
for -measurable , we have ((())) =, i.e. the conditional expectation () is in the sense of the L 2 (P) scalar product the orthogonal projection from to the linear subspace of -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem .)
If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since (/ >) = (>) = = = [()] [] =. More generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate ...