Calculations with Normal Distributions (Edexcel IAL Maths)

Throughout this section we will use the random variable $X \sim \mathrm{N} ( \\mu , \\sigma ^{2} )$ . 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

Spec point — 5.1

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 $P(a\leq X\leq b)=P(a<X<b)$ 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 $X \sim \mathrm{N} ( \\mu , \\sigma ^{2} )$ can be coded to model the standard normal distribution equation using the formula

$Z=\frac{X-μ}{σ}$

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

• 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
  • $P(X=x)=0$

• $P(X<μ)=P(X>μ)=0.5$

  • 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 $P(X=a)=0$ you can use:

  • $P(X<a)+P(X>a)=1$
  • P (X greater than a)=1 - P (X less than a)=1 - capital φ ((a - μ)/(σ))
  • P (a less than X less than b)=P (X less than b) - P (X less than a)=capital φ ((b - μ)/(σ)) - capital φ ((a - μ)/(σ))

Worked Example The random variable $X \sim \mathrm{N} ( 20 , 5^{2} )$. Calculate:

(a) $P(X\leq22)$,

(b) $P(18\leq X\leq27)$

Answer:

Inverse Normal Distribution

Spec point — 5.1

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 $P(X<a)$ is more than 0.5 or $P(X>a)$ is less than 0.5 then a should be bigger than the mean

    • z will be positive

  • If $P(X<a)$ is less than 0.5 or $P(X>a)$ 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 value substitute it into the formula****$z=\frac{a-μ}{σ}$ 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 $P(X<μ+a)$ will be $( \frac{100 + \\alpha }{2} ) \%$

  • 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 $z=\frac{(μ+a)-μ}{σ}=\frac{a}{σ}$

Worked Example The random variable $W \sim \mathrm{N} ( 50 , 36 )$.

Find the value of $w$ such that $P(W>w)=0.7673$

Answer:

Exam Hint

• The most common mistake students make when finding values from given probabilities is forgetting to check whether the *z-*value should be negative or not. Avoid this by checking early on using a sketch whether z is positive or negative and writing a note to yourself before starting the other calculations.