A random variable has a continuous uniform distribution if the probability that it takes on a value, within a finite interval [a, b], is the same for any sub-interval of equal length.
This distribution is analogous to the discrete uniform distribution, which assigned the same probability to each result of the random experiment, but in this case the variable to be considered is continuous. For example, the experiment that consists of selecting a real number at random, between the values a and b, follows the uniform distribution. Here is its graph:
In mathematical notation, the continuous uniform distribution has a density function defined as a piecewise or piecewise function, which can be written as:
The graph of this function, known as density curve or function, is a rectangle, therefore the continuous uniform distribution is also known as rectangular layout y is the simplest of the continuous distributions.
The area under the graph of a probability distribution is equal to 1 and always takes positive values. The uniform distribution meets these criteria. It is not necessary to integrate directly to check that the area is 1, since the area of the shaded rectangle in Figure 1 can be calculated using the formula:
Area = base x height = (b − a) x [1 / (b − a)] = 1
Knowing the area under the density curve is very important, because there is a relationship between the area and the probability of occurrence of an event, which for this distribution is determined in the next section.
The continuous uniform distribution is characterized by its:
Let X be the continuous random variable, which belongs to the interval [a, b], then:
The distribution function calculates the probability that the random variable X takes a value x from among the possible values of the interval [a, b]. For a continuous distribution, it is generally calculated like this:
In the case of the continuous uniform distribution, said probability F (x) equals the area of the rectangle whose base is (x-a) and its height is (b-a):
Mathematically, if F (x) = Pr (X = x) the following function is established in parts, according to the previous result:
In this way, what has been said before is verified: the probability only depends on the value of (x-a) and not on its location in the interval [a, b]. The graph of the distribution function is:
After doing numerous experiments with the continuous random variable, its average value is called expected value, is denoted as E (X) and is calculated by the following integral:
V (X) = E (Xtwo) - E (X)two
Therefore:
D (X) = √ V (X)
It can easily be verified that the median, which is the central value of the uniform distribution, is equal to the mean, and since there is no one value that repeats more than others, since all are equally probable in the interval [a, b ], fashion does not exist.
Regarding symmetry, the uniform distribution is symmetric and the kurtosis, which is the degree to which the values are concentrated around the center, is -6/5.
Various situations can be modeled through continuous distribution, and thus their behavior can be predicted. Here are some examples:
A company that provides electrical service provides levels of voltages uniformly distributed, between 123.0 V and 125.0 V. This means that in the domestic outlet it is possible to obtain any voltage value that belongs to that range..
So, as seen above, the graph of the density function is the rectangle in red:
Calculating the probability of having a voltage within the given interval is very easy, for example, what is the probability that the company sends a voltage lower than 123.5 V?
This probability equals the area of the rectangle shaded in blue:
P (X<123.5) = (123.5 −123.0)x 0.5 = 0.25
And what is the probability that the voltage delivered is greater than 124.0 V?
Since the total area is equal to 1, the probability sought is:
P (X> 124.0 V) = 1 - (1 × 0.5) = 0.5
It makes sense, since 124.0 is precisely the value in the center of the interval.
A certain random variable X has a uniform distribution in the interval [0,100]. Decide:
a) The probability that the value of X is less than 22.
b) The probability that X takes values between 20 and 35.
c) The expected value, variance and standard deviation of this distribution.
It is determined in a similar way to the previous example, but first we must determine the height of the rectangle, remembering that the total area must be equal to 1:
Area = 100 × height = 1
Therefore the rectangle has a height equal to 1/100 = 0.01
P (X<22) = 22×0.01 = 0.22
The requested probability equals the area of the rectangle whose width is (35 - 20) and whose height is 0.01:
P (22 If you prefer to go directly to the distribution function given above, then you just have to substitute the values in: P (20≤X≤35) = F (35) -F (20) With F (x) given by: F (x) = (x-a) / (b-a) The values to enter are: a = 0 b = 100 F (35) = (35-0) / (100-0) = 0.35 F (20) = (20-0) / (100-0) = 0.20 P (20≤X≤35) = 0.35-0.20 = 0.15 The expected value is: E (X) = (a + b) / 2 = (100 + 0) / 2 = 50 The variance is: V (X) = (b-a)two/ 12 = (100-0)two/ 12 = 833.33 And the standard deviation is: D (X) = √833.33 = 28.87 This distribution is useful when carrying out statistical simulation processes or when working on events whose frequency of occurrence is regular.. Some programming languages generate random numbers between 0 and 1, and as can be seen from the previous examples, the probability distribution followed is uniform. In this case, the interval to consider is [0,1]. If you have an experiment in which the events have regularity, as explained before, you can, in principle, assign each one the same probability of occurrence. In this case, the probabilistic model of uniform distribution provides information for the analysis.. The uniform distribution is also used in rounding the differences between the observed values and the real values of a variable, assuming a uniform distribution of the error in a given interval, according to rounding, usually from -0.5 to +0.5.Answer c
Applications
Random number generation
Sampling of arbitrary distributions
Rounding off errors
References
Yet No Comments