Normal random variable

Short description and context

Normal random variable

The normal random variable is one of the most important random variables in statistics. Its particular importance is due to the implications of the central limit theorem. The probability density function for the normal distribution is: $$ f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ where $\mu$ and $\sigma$ are parameters. The expectation and variance for this random variable are $\mathbb{E}(X) = \mu$ and $\textrm{var}(X) = \sigma^2$

Syllabus Aims

  • You should be able to explain why the normal random variable is used so often when interpretting the outcome of experiments by making reference to the central limit theorem.
  • You should be able to write out the probability density function for a normal random variable.
  • You should be able to prove that the probability density function for a normal random variable is properly normalised.
  • You should be able to obtain expressions for the expectation of a random variable by using integration and arguments based on the symmetry of the probability density.
  • You should be able to obtain an expression for the variance of a normal random variable by means of integration by substitution.

Extending your understanding

Description and link

Module

Author

Writing a short project on normal random variables that includes some small programming components. SOR3012 G. Tribello

Contact Details

School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN

Email: g.tribello@qub.ac.uk
Website: mywebsite