Search results
Results From The WOW.Com Content Network
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the ...
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by E [ X ] = 1 λ . {\displaystyle \operatorname {E} [X]={\frac {1}{\lambda }}.} In light of the examples given below , this makes sense; a person who receives an average of two telephone calls per hour can expect that the time ...
For the alternative formulation, where X is the number of trials up to and including the first success, the expected value is E(X) = 1/p = 1/0.1 = 10.
Conditional expectation. In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can ...
The variance of a random variable is the expected value of the squared deviation from the mean of , : This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:
About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and n p ≤ 10. [31] Letting and be the respective cumulative density functions of the binomial and Poisson distributions, one has:
where μ is the expected value of the random variables, σ equals their distribution's standard deviation divided by n 1 ⁄ 2, and n is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant .