Properties of a continuous uniform distribution

This article continues from the previous article. It may be a good idea to go through it before attempting this. This article explores properties of a continuous uniform distribution. We shall look at the variance, mean and cumulative distribution.
Given that the continuous random variable X has probability density function (p.d.f)
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We shall find;

• E(X)
• Var(X)
• the cumulative distribution function F(x)

For E(X) imagine;
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…then by symmetry;
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Now we shall find the formula for variance Var(X). To find Var(X)we shall use the continuous version of variance.
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Note that we could also use E(X²) – (E(X))² this will be much difficult so we shall use the one above it.
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Using…
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…we get…
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Note that;
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…so we have…
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So the formula for the variance of the continuous uniform distribution is;
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Now we shall find the cumulative distribution function F(x). If a≤x≤b, then;
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Key Points

So the key points for the properties of a continuous uniform distribution are;

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• IMAGE
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Below we shall look at some examples;

Example

The continuous random variable Y is uniformly distributed over the interval [4, 7]. Find;

• E(X)
• Var(X)
• the cumulative distribution function of X.

For E(X), we know for the derived properties that;
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…therefore…
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Now we shall find Var(X). We know from above that;
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…so…
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To find the cumulative distribution function of X, F(x); The given interval is [4, 7] so we replace a with 4 and b with 7 to get;
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…therefore the cumulative distribution function becomes;
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Example

The continuous random variable Y is uniformly distributed over the interval [a, b]. Given E(Y) = 1 and Var(X) = 16/3, Find the value of a and the value of b.
In this example we shall take advantage of the E(Y) and Var(Y) expressions. So we shall form two expressions from E(Y) and Var(Y). For E(Y);
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For Var (Y);
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So we shall solve the equations of E(X) and Var(X) simultaneously;
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So for (2 – 2a) = 8
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…for(2 – 2a) = -8
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So here we shall use the values in the first part since a is less than b. The answer is;
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Example

The continuous random variable X is uniformly distributed over the interval [-3, 5].

• First we shall find E(X)
• and use integration to find the value of X

First of all E(X) = 1.
To find Var(X) we integrate in the given range.
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