Continuous uniform/rectangular distribution
This article explores the continuous uniform/rectangular distribution. Prior knowledge of continuous random variables will be useful here.
The continuous random variable X with probability distribution function (p.d.f);
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where a and b are constants is called a continuous uniform (rectangular) distribution.
The distribution is denoted by;
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Below is the sketch;
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Below we shall look at some examples;
Example
In this example the continuous random variable X ~ [a, b]
- First we will write down the distribution of Y = 5X – 4
- then find P(3.2 < X < 4.3)
To find the distribution of Y=5X – 4 we substitute the lower and upper limits of 3 and 5 into Y = 5X – 4, so we shall do it for the lower limit first.
Lower limit
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Upper limit
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Therefore the distribution of Y = 5X – 4 is;
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Now we shall find P(3.2 < X < 4.3). Here we’re trying to find the area of the rectangle in the interval x=3.2 and x=4.3. It is useful to sketch the probability density function (p.d.f) first.
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p(3.2 < x < 4.3) is area of the shaded section on the sketch;
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...therefore...
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Example
This continuous random variable X has probability density function (p.d.f) as shown in the following diagram;
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We shall find;
- the value of k
- and P(3 < X < 3.5)
To find the value k, we consider the fact that the area of the rectangle has to equal to 1. So;
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…we solve to find the value of X;
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Next we shall find P(3 < X < 3.5). Here we simply multiply the rectangle by f(x), that is;
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