Uniform - Uniform (or Rectangular) Distribution: Among the...

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Uniform (or Rectangular) Distribution: Among the continuous probability distribution, the uniform distribution is the simplest one of all. The following figure shows an example of a uniform distribution. In a uniform distribution, the area under the curve is equal to the product of the length and the height of the rectangle and equals to one. Figure 1 where: a=lower limit of the range or interval, and b=upper limit of the range or interval. Note that in the above graph, since area of the rectangle = (length)(height) =1, and since length = (b - a), thus we can write: (b - a)(height) = 1 or height = f(X) = 1/(b - a). The following equations are used to find the mean and standard deviation of a uniform distribution:
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Example: There are many cases in which we may be able to apply the uniform distribution. As an example, suppose that the research department of a steel factory believes that one of the company's rolling machines is producing sheets of steel of different thickness. The thickness is a uniform random variable with values between 150 and 200 millimeters. Any sheets less than 160 millimeters thick must be scrapped because they are unacceptable to the buyers. We want to calculate the mean and the standard deviation of the X (the tickness of the sheet produced by this machine), and the fraction of steel sheet produced by this machine that have to be scrapped. The following figure displays the uniform distribution for this example. Figure 2 Note that for continuous distribution, probability is calculated by finding the area under the function over a specific interval. In other words, for continuous distributions, there is no probability at any one point. The probability of X>= b or of X<= a is zero because there is no area above b or below a, and area between a and b is equal to one, see figure 1. The probability of the variables falling between any two points, such as c and d in figure 2, are calculated as follows: P (c <= x <="d)" c)/(b a))=? In this example c=a=150, d=160, and b=200, therefore: Mean = (a + b)/2 = (150 + 200)/2 = 175 millimeters, standard deviation is the square root of 208.3, which is equal to 14.43 millimeters, and P(c <= x <="d)" 150)/(200 150)="1/5" thus, of all the sheets made by this machine, 20% of the production must be scrapped.)=. ... Normal Distribution or Normal Curve:
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Normal distribution is probably one of the most important and widely used continuous distribution. It is known as a normal random variable, and its probability distribution is called a normal distribution. The following are the characteristics of the normal distribution: Characteristics of the Normal Distribution: 1. It is bell shaped and is symmetrical about its mean. 2
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Uniform - Uniform (or Rectangular) Distribution: Among the...

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