Continuous random variables

This section explores random variables. By the end of this entry you should be able to decide whether something can be a probability density function, find the cumulative distribution function given the probability density function, find the probability density function given the cumulative distribution function, find the mean and variance of a random variable using its probability density function, find the mode, mean of a random variable using its probability density function. Below are the key points covered in this entry;

• For a continuous random variable X
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• Cumulative distribution function f(x)
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• • The median m satisfies F(M) = 0.5
  • The lower Quartile Q_{1 satisfies F(Q1) = 0.25}
  • The upper Quartile Q₃ satisfies F(Q₃) = 0.75
• The mode is the x value at the highest point of the function.

First let us look at the concept of a continuous random variable and its probability density function;
The histogram below shows the variable t, the time in seconds taken by 100 contestants to complete their meal in a food contest.
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In the histogram above we know that the area of a bar = k x total frequency. Let k = 1 above. If we let the total frequency = n and let k=1/n the;
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We know that the total probability is always equal to 1. If we adjust the scale of the histogram accordingly we get;
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In the new histogram the total area is now equal to 1. If a contestant is chosen at random, then the probability of getting a value between any two times is now the area of those two times. For instance the probability of the contestant taking a time between 110 and 130 seconds is;
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Above we have drawn a probability density function f(x) in the red line. It is a smooth version of the histogram. Below we shall explore the important properties of a probability density function (p.d.f).
If X is a continuous random variable with p.d.f f(x) then;

• f(x)>0 since we cannot have negative probabilities
• P(a < x
• Below area = 1, since the total area under the curve = 1
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Below we shall look at some examples using the properties shown above.

Determining probability density functions

Below we shall look at what functions