Calculations with Normal Distributions (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Revision Note

Normal Distribution - Calculations

Throughout this section we will use the random variable $X~\mathrm{N}(\mu ,{\sigma }^2)$ . For normal, $X$ can take any real number. Therefore any values mentioned in this section will be assumed to be any real number.

Calculating Normal Probabilities

How do I find probabilities using a normal distribution?

• The area under a normal curve between the points $x=a$ and $x=b$ is equal to the probability P(a < X < b )

  • Remember for a normal distribution $P(a\le X\le b)=P(a<X<b)$ so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)

• The equation of a normal distribution curve is complicated so the area must be calculated numerically

• You will be expected to standardise all normal distributions to $z$ and use the table of the normal distribution to find the probabilities

  • It is likely that your calculator has a function that can find normal probabilities, if so it is a good idea to learn to use it so that you can check your probabilities

  • However you must show your calculations to get the z values and use the tables to get all the marks

How do I calculate the probability for a normal distribution?

• A random variable $X~\mathrm{N}(\mu ,{\sigma }^2)$  can be coded to model the standard normal distribution $Z~\mathrm{N}(0,1^2)$ using the formula

$Z=\frac{X-\mu }{\sigma }$

• You can calculate a probability $\mathrm{P}(X<x)$ using the relationship $\mathrm{P}(X<x)=\mathrm{P}\left(Z<\frac{x-\mu }{\sigma }\right)$

• Always sketch a quick diagram to visualise which area you are looking for

• Once you have determined the z value use the table of the normal distribution to find the probability

  • Refer to your sketch to decide if you need to subtract the probability from one

The probability of a single value is always zero for a normal distribution

• You can picture this as the area of a single line is zero

  • $\mathbf{P}\mathbf{(}\mathit{X}\mathbf{=}\mathit{x}\mathbf{)}\mathbf{=}\mathbf{0}$

• $\mathrm{P}(X<\mu )=\mathrm{P}(X>\mu )=0.5$

  • You can look at which side of the mean x is on and the direction of the inequality to decide if your answer should be greater or less than 0.5

• As $\mathrm{P}(X=a)=0$ you can use:

  • $\mathrm{P}(X<a)+\mathrm{P}(X>a)=1$

  • $\mathrm{P}(X>a)=1-\mathrm{P}(X<a)=1-\mathrm{\Phi }\left(\frac{a-\mu }{\sigma }\right)$

  • $\mathrm{P}(a<X<b)=\mathrm{P}(X<b)-\mathrm{P}(X<a)=\mathrm{\Phi }\left(\frac{b-\mu }{\sigma }\right)-\mathrm{\Phi }\left(\frac{a-\mu }{\sigma }\right)$

Inverse Normal Distribution

Given the value of P(X < a)  or P(X > a)  how do I find the value of a?

• Given a probability you will have to look through the table of the normal distribution to locate the z-value that corresponds with that probability

• Look at whether your probability is greater or less than 0.5 and the direction of the inequality to determine whether your z-value will be positive or negative

  • If $\mathrm{P}(X<a)$ is more than 0.5 or $\mathrm{P}(X>a)$ is less than 0.5 then a should be bigger than the mean

    • z will be positive

  • If $\mathrm{P}(X<a)$ is less than 0.5 or $\mathrm{P}(X>a)$ is more than 0.5 then a  should be smaller than the mean

    • z will be negative

• You do not need to remember these, a sketch will help you see it

  • Always sketch a diagram

• If your probability is less than 0.5 you will need to subtract it from one to find the corresponding z value

  • Remember that the position of the z-value will not change, only the direction of the inequality

• Once you have the correct $z$ value substitute it into the formula $z=\frac{a-\mu }{\sigma }$   and solve to find the value of a

• Always check that your answer makes sense by considering where a is in relation to the mean

Given the value of P(µ- a < X < µ + a) I find the value of a  ?

• A sketch making use of the symmetry of the graph is essential

• If you are given $P(\mu -a<X<\mu +a)=\alpha \%$  then $\mathrm{P}(X<\mu +a)$ will be $\left(\frac{100+\alpha }{2}\right)\%$ 

  • This is easier to see from a sketch than to remember

  • You can then look through the tables for the corresponding z-value and substitute into the formula  $z=\frac{(\mu +a)-\mu }{\sigma }=\frac{a}{\sigma }$