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The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions. The mode is not necessarily unique in a given discrete distribution since the probability mass function may take the same maximum value at several points x 1 , x 2 , etc.
The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1]. The logit-normal distribution on (0,1). The Dirac delta function , although not strictly a probability distribution, is a limiting form of many continuous probability functions.
The median of a normal distribution with mean μ and variance σ 2 is μ. In fact, for a normal distribution, mean = median = mode. The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean. The median of a Cauchy distribution with location parameter x 0 and scale parameter y is x 0, the location parameter.
In statistics, a multimodal distribution is a probability distribution with more than one mode (i.e., more than one local peak of the distribution). These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions.
Let and be respectively the cumulative probability distribution function and the probability density function of the ( , ) standard normal distribution, then we have that [2] [4] the probability density function of the log-normal distribution is given by:
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.The general form of its probability density function is [2] [3] = ().
If the distribution is symmetric, then the mean is equal to the median, and the distribution has zero skewness. [3] If the distribution is both symmetric and unimodal, then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4,...
Moreover, when =, the bound is equal to /, which is the maximum distance between the median and the mean of a unimodal distribution. A similar relation holds between the median and the mode θ: they lie within 3 1/2 ≈ 1.732 standard deviations of each other: