Choosing a distribution and approximationsEdexcel A-Level Maths: Revision notes

$X\sim B(n,p)$ is suitable when all of these hold:

• a fixed number $n$ of trials;
• each trial has two outcomes (success or failure);
• trials are independent;
• the probability of success $p$ is constant. The context must support these. Ten seeds each germinating with probability 0.45, independently, is binomial. State the conditions in context, for example "each seed germinates independently of the others, with the same probability".

The model can fail when:

• Independence fails: students in the same tutor group live in the same roads, or family members share a habit.
• The probability changes: sampling without replacement from a small population changes the probabilities for each draw (drawing 6 cards from 10 without replacement).
• The number of trials is not fixed: counting until something happens.
• Selection is not random: choosing a convenient group rather than a random sample of the population. To make a binomial model more suitable, take a random sample from the whole population so that trials are independent with the same $p$.

The Normal distribution suits a continuous variable that is symmetrical and bell-shaped, such as heights or masses. It is a poor model when:

• the data are skewed (for example incomes, or times with a long upper tail);
• the variable is bounded or discrete with only a few values (the number of sixes in 5 rolls);
• the distribution has more than one peak. A histogram of the data that is roughly symmetrical about its mean with most values within about two standard deviations supports using a Normal model.

If $X\sim B(n,p)$ with $n$ large and $p$ close to 0.5, then $X$ can be approximated by $Y\sim N\big(np,\;np(1-p)\big).$ Use the mean $np$ and the variance $np(1-p)$ (not its square root). For $X\sim B(60,0.45)$: $Y\sim N(27,14.85)$. The approximation works because the binomial distribution is roughly symmetrical when $p$ is near 0.5. It becomes poor when $p$ is close to 0 or 1 or when $np$ is small: for $B(50,0.05)$ it gives positive probability to negative values.

A binomial variable takes integer values (discrete) but the Normal distribution is continuous. The continuity correction treats each integer $k$ as the interval from $k-0.5$ to $k+0.5$:

• $P(X\le k)\approx P(Y<k+0.5)$
• $P(X\ge k)\approx P(Y>k-0.5)$
• $P(X<k)=P(X\le k-1)\approx P(Y<k-0.5)$
• $P(X>k)=P(X\ge k+1)\approx P(Y>k+0.5)$
• $P(a\le X\le b)\approx P(a-0.5<Y<b+0.5)$ Example: $X\sim B(80,0.55)$, $Y\sim N(44,19.8)$. $P(40\le X\le48)\approx P(39.5<Y<48.5)=0.688$.