Continuous Random Variables

Continuous Random Variables

A continuous random variable is one that can take any real number in an interval, rather than a discrete probability that we have looked at previously. A good example of this is a long jumper, who for each jump can get a range of values.

Probability Density Function

Usually continuous functions are modelled by a probability density function. We find the probability of this function by taking the integral over the range of that we want to find the probability


If we integrate the function over the whole range then it should equal one.

If we want to find the probability a long jumper will leap between 4.5m and 4.6m we take the integral of a probability of this function over that range.


Practice Exam Example

Question 7 from the 2010 exam 1

prob continuous Question 2

A probability density function must have a total area under it equal to one.


Step 1 Expand the function:


Step 2 integrate it using our integral power rule


Step 3 Substitute in the bounds and solve for x

a(\frac{125}{2}-\frac{125}{3})=1 \newline a(\frac{125}{6}=1 \newline a=\frac{6}{125}

Continuous random variables rely on us knowing our integration rules, so if ensure you know them before starting on this section.