Uniform Distribution as the Limiting Form of a Density Function

The uniform distribution, denoted by 𝑈(𝑥; 𝐴, 𝐵) = 1/(𝐵 − 𝐴) if 0 < 𝐴 < 𝑥 < 𝐵 < ∞ and zero otherwise, is the simplest probability density functions of a continuous random variable X . For a continuous random variable X on the interval (0, 1), a three parameters density function, denoted by ℎ(𝑥; 𝐴, 𝐵, 𝑛) , is constructed so that its limiting form is the uniform density function 𝑈(𝑥; 𝐴, 𝐵) in which 𝑛 → ∞ .


Introduction
The intention is to generalize the well-known uniform distribution ( ; , ) where 0 < < < < ∞. The power functions and 1/ for all ∈ (1, ∞), are used to construct a probability density function, denoted by ℎ( ; ), whose limiting form, in which → ∞, is the uniform distribution ( ; 0, 1), with 0 < < 1. This density function will be called the standard h-distribution. A general form of ℎ( ; ), denoted by ℎ( ; , , ), is also constructed so that its limiting form is the uniform distribution ( ; , ) in which → ∞. The density function ℎ( ; ) and its general form are not only important as generalization of an existing distribution, but also, they are applicable in many branches of science, such as the medical field, mineral industry, and technology. First, let us recall the following well-known facts that are needed for this work.
The set of all possible outcomes of a statistical experiment is called a sample space. A sample space is called continuous if it contains a noncountable number of possibilities. A random variable is a real-valued function on the sample space. When the random variable assumes noncountable number of values, it is called a continuous random variable. To calculate the probabilities that a continuous random variable assumes values from a certain interval of real numbers, we must derive its probability density function. The probability density function for a continuous random variable is constructed so that the area under its curve bounded by the x-axis is equal to 1. A function ( ) is called a probability density function of a continuous random variable X, if the following hold.
For a continuous random variable, nonzero probabilities are associated only with interval of numbers. Because of condition 3 in (0), the probability of occurrence of a single value of this random variable is zero.
The cumulative distribution function, the rth moment of the distribution, and the variance of a continuous random variable with probability density function ( ), are denoted, respectively, by Standard deviation √ ( ) is the amount of dispersion from the mean of values that random variable can assume.
the points = 0 and = 1. The area between the two curves is The function denoted by is a probability density function for a continuous random variable (defined over a set of real numbers) due to 1. ℎ( ; ) ≥ 0 for all real numbers x, since for > 1, 0 < < 1, the ratio  (2) 3.
In the medical field, it is reasonable to claim that the time it takes to complete a surgery on part of a body is proportional to the size of the wound that is needed to perform the surgery through. The diameter of the needle that is needed to sew the wound is also proportional to the size of the wound. Hence, the diameter of the needle that is needed is, consequently, proportional to the time that is needed to sew the wound and complete the surgery. So, the diameter of a certain needle may be taken to be a random variable that is distributed uniformly between, say, 0 and 1 units of time.
Proof. For any fixed in (0, 1), lim →∞ 1/ = 0 = 1 and lim →∞ = 0. Therefore, In piecewise defined form, which is indeed the uniform distribution ( ; 0, 1) of the random variable X over the interval [0, 1]. ∎ Figure 1.1 below shows how the standard h-distribution ℎ( ; ) approaches the uniform distribution ( ; 0, 1) as approaches infinity. It shows that, as is getting bigger, the graph of ℎ( ; ) is getting closer to the graph of the density function ( ; 0, 1). For each > 1, the density function ℎ( ; ) is skewed to the right and part of the graph is above the horizontal line = 1. As is getting bigger, the left part of the curve ℎ( ; ) is getting closer to the vertical line = 0, the right part of the curve is getting closer to the vertical line = 1, and the part of the curve that is above the line = 1 is getting closer to this line. Only at infinity, the density function ℎ( ; ) equals the uniform distribution ( ; 0, 1). The curve ℎ( ; ) has a bell shape. But due to a lack of symmetry and the heavy right skewness, its median that is very sensitive to extremely small or extremely large values, is greater than its mean for > 1. However, the mean and mode of ℎ( ; ) must coincide at one value of the parameter . See Proposition 6 below.

Proof. lim →∞
The value of , which maximizes the density function ℎ( , ) over its domain, is called the mode or modal value.
The chart also shows that the mode of ℎ( , ) is smaller than its mean if = 3 and greater than its mean if = 4.

Proposition 6
The mode and mean of the density ℎ( , ) are the same at one value of inside the interval (3, 4).
Proof. The mode and the mean of the curve ℎ( , ) are characterized by . For a real number greater than 1, the is clearly continuous. Since That is, at ≈ 3.54838, the mode and the mean of the curve ℎ( , ) are equal.

A General Form of the Standard h-Distribution ℎ( ; ).
A probability density function with three parameters , and n can be defined for a continuous random variable X, by The function (9) is a probability density function on ( , ) due to the following: Back to a surgery performed on somebody. The initial wound will be on the surface of the body followed by a smaller one on an interior organ inside the body at which the main operation is taking place. That is how the numbers A and B came to the picture.
a. The cumulative distribution of the random variable X with the probability density function ( ; , , ).
Indeed, ( ), ( ), and ( ) are the cumulative distribution function, expected value, and variance, of the random variable X with the uniform distribution ( ; , ), respectively. ∎

Conclusion
The density function ℎ( ; , , ) that is constructed in this paper, is a generalization of the well-known uniform distributions ( ; , ), ℎ 0 < < < < 1. It has applications in several physical phenomena including the medical field, minerals industry, and technology. When the parameter , on its way up, fall in the closed interval [3 ,4], the density ℎ( ; , , ) is approximated by the normal distribution. So, the density ℎ( ; , , ) is a good addition to those densities that may be approximated by the normal distribution under certain conditions such as the binomial, the hypergeometric, the Poisson, and the gamma. But only when → ∞ will the distribution ℎ( ; , , ) equals the uniform distribution U(x; A, B) , where 0 < < < < 1 . Thus, the uniform curve ( ; , ) is the curve ℎ( ; , , ) with equals ∞. If the mode ( ) = 2 /(1− 2 ) of ℎ( ; ) is given, then the power can be recovered and the distribution ℎ( ; ) can appear. The discussion of properties of the standard h-distribution ℎ( ; ) is as follows.
2. There are infinitely many standard h-distributions ℎ( ; ), one for every value of the parameter .
3. For each > 1, the curve ℎ( ; ) has two parts: one above the line = 1, and the other below the same line.
5. The density function ℎ( , ) and its mode, mean, and maximum height, are characterized by the parameter n.

Its mean
is an increasing sequence that converges to 1 2 , which is the mean of ( ; 0, 1).
8. The curve ℎ( ; ) peaks at only one value 0 of the parameter that is inside the open interval (3, 4).
9. If < 0 , the mode of ℎ( ; ) is smaller than its mean. If > 0 , its mode is greater than its mean.
10. For ∈ [3, 4], its mode ( ) and mean , are very close to each other. That is, ℎ( ; ) is very close to the normal distribution.