Continuous Random Variables

A random variable is continuous if the values can be expressed as an interval. Calculations are for continuous variables, for example heights or times, that is, something that can be measured rather than counted. In order to calculate probabilities, we need to find the area under graphs, so it is important that you are confident with integration techniques before working through this page. In the exam, questions are usually the long part B questions. Typical questions are given in the exam-style questions below.


Key Concepts

On this page, you should learn about

  • continuous random variables and their probability density functions
  • mode of continuous random variables
  • median of continuous random variables
  • mean of continuous random variables
  • variance of continuous random variables

Summary

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Test Yourself

Here is a quiz that gets you to check the initial conditions for continuous random variables


START QUIZ!

Here is a quiz about finding the mean, mode and median from a continuous random variable


START QUIZ!

Exam-style Questions

Question 1

The probability density function of X is given by

\(f\left(x\right)=\left\{\begin{matrix}ax^n\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<1\\otherwise\\\end{matrix}\)

a) Show that a = n + 1

b) Given that E(X) = 0.75, find a and n


Hint

Full Solution

 

Question 2

The continuous random variable X has a probability density function given by

\(f\left(x\right)=\left\{\begin{matrix}a\\asin\left(\frac{\pi x}{4}\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<2\\2\le x<4\\otherwise\\\end{matrix}\)

a) Draw a sketch of f

b) Show that the value of \(a=\frac{\pi}{2\pi+4}\)

c) Find E(X)

d) Find the exact value of the median of X

e) Find \(P(X\le2|X\le3)\)


Hint

Full Solution

 

Question 3

The continuous random variable X has a probability density function given by

\(f\left(x\right)=\left\{\begin{matrix}k\bullet a r c o s\left(x\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}-1\le x<1\\otherwise\\\end{matrix}\)

a) Draw a sketch of f

b) State the mode of X

c) Find \(\int a r c o s\left(x\right)dx\)

d) Find E(X)

e) Find Var(X)


Hint

Full Solution

 

MY PROGRESS

How much of Continuous Random Variables have you understood?