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To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals ...
The mean is the probability mass centre, that is, the first moment. The median is the preimage F −1 (1/2). The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by [] =.
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\displaystyle f} , mean μ {\displaystyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
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 only take on a subset of ...
Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. [2] [3] Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.
The inverse of the harmonic mean (H X) of a distribution with random variable X is the arithmetic mean of 1/X, or, equivalently, its expected value. Therefore, the harmonic mean (H X) of a beta distribution with shape parameters α and β is:
The theory of median-unbiased estimators was revived by George W. Brown in 1947: [8]. An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates.
By the central limit theorem, because the chi-squared distribution is the sum of independent random variables with finite mean and variance, it converges to a normal distribution for large . For many practical purposes, for k > 50 {\displaystyle k>50} the distribution is sufficiently close to a normal distribution , so the difference is ...