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

$Pr(a<X<b)=\int\limits_{a}^{b}{f(x)}dx$

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.

$Pr(4.5<X<4.6)=\int\limits_{4.5}^{4.6}{f(x)}dx$

Practice Exam Example

Question 7 from the 2010 exam 1

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

$\int\limits_{0}^{5}{f(x)}dx=1$

Step 1 Expand the function:

$\int\limits_{0}^{5}{5ax-ax^{2}}dx=1$

Step 2 integrate it using our integral power rule

$[\frac{5ax^{2}}{2}-\frac{ax^{3}}{3}]_{0}^{5}=1$

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.

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