Probability density mean and distribution

This article is a continuation from the previous random variables article. This article covers mean and variance of a probability density function.
First we shall discuss a few key points before actually looking at some examples. We shall start with the mean. You may know that the mean for a discrete distribution is;
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For the mean of a continuous distribution we simply replace ∑ with;
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…and p(x) by f(x)dx, therefore for the mean of probability density function we get;
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For the variance we consider the variance of a discrete distribution;
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For the variance of a continuous distribution again we replace ∑ with;
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…and p(x) with f(x)dx.
So the variance of a probability density function is;
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You may have noticed from the equation above that E(x) ² is;
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Now we shall look at some examples;

Example

Consider a random variable Y having a probability density function;
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In this example we’re going to find E(Y), Var(Y), E(2Y-3) and Var(2Y-3). We shall start with E(Y); As we saw above;
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Here;
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So we get;
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We shall now find the variance Var(Y); Using the variance formula;
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We have already found µ (mean) above. We need to form an expression for x²f(x) for the formula as we did above;
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…so the variance becomes;
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We shall find E(2Y – 3), In discrete distribution we use saw that;
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The formula can also apply to the continuous variables. So we have;
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For E(2Y – 3) we do the same to get;
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Example

In this example suppose a random variable X has probability density function;
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The sketch of the p.d.f will look like the following;
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We can actually find E(X) from the graph. E(X) is the line of symmetry. This has been shown above. Therefore;
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How about P(X > µ); We’re trying to find P(X > E(µ)) so we have;
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Example

In this example suppose the random variable X has probability density function;
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Let us find E(X) and Var(X).
For E(X); We could use the sketch of the p.d.f shown below;
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However the graph above shows us that it is not symmetrical, in this case we will need to integrate to find E(X) as follows;
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For Var(X);
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