Cumulative distribution function

This article is a continuation from the previous continuous random variables article. It may be a good idea to read through it before attempting this section. This article covers cumulative distribution for continuous random variables. Knowledge of the cumulative distribution, F(x) = P(X ), for a discrete random variable will be very useful here.
Consider the continuous random variable T such that F(t) = f(x) the for the continuous random variable X;
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F(x) is simply the area below the curve as shown below;
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Notice that for the cumulative distribution function (c.d.f) we use the notation F(x) (with capital F) and f(x) for the probability density function (p.d.f)
If X is a continuous random variable with cumulative distribution function (c.d.f) F(x) and probability density function p.d.f f(x) the;
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So we integrate the probability density function to find the cumulative distribution function (c.d.f) and vice versa; using differentiation;
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We shall explore some examples below;

Example

Given that the random variable X has probability density function;
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Let us find F(x).
Here we simply integrate the function between 1 and x. Note also from the function f(x) is zero between – ∞ and 1. We’re going to use a lower bound of one. From the given interval, so we have;
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We have integrated between 1 and x. F(x) = 1 when x=3. So we use an indefinite integral and find C.
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…when x=3 F(x)=1, therefore;
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Therefore the cumulative distribution function becomes;
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Notice in our cumulative distribution function above we have defined F(x) for the entire range between -∞ and ∞. You can see that F(x) = 0 for all values less than 1 and F(x)=1 for all values greater than 3.
We can plot the function on a diagram as shown below;
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Example

Here is another example. The random variable X has probability density function, where;
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Let us find F(x), the cumulative distribution function; Again here we could find the function in two ways.
We know that if x ≤ 1 F(x) = 0, therefore F(1)=0 and where x is between 1 and 2 i.e 1Example

How about supposing the random variable x has cumulative distribution function;
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Now suppose we had to find P(x ≤ 1.5). We use F(x) = P(X = x) so we get;
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For P(0.5 ≤ x ≤1.5) we know that;
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…so we get;
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It is very important to remember that the probability of a single value happening in a continuous distribution is always 0, for example;
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Now how about finding the probability density function f(x). If we differentiate F(x) we get;
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So the probability density function f(x) is;
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