9709 · 5.5
The normal distribution flashcards
Revision flashcards for Cambridge 9709 The normal distribution (syllabus 5.5). Flip, recall, then mark a real past-paper question.
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What does the notation $X \sim N(\mu, \sigma^2)$ signify?
It means that the random variable $X$ follows a normal distribution with a mean of $\mu$ and a variance of $\sigma^2$.
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What are the key properties of a normal distribution curve?
It is a bell-shaped, continuous curve, perfectly symmetric about the mean $\mu$. The mean, median, and mode are all equal. The total area under the curve is 1.
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What is the standard normal distribution?
A special case of the normal distribution with a mean of 0 and a variance of 1. It is denoted by $Z \sim N(0, 1^2)$.
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What is the formula for standardising a normal variable $X$?
$Z = \frac{X - \mu}{\sigma}$. This transformation converts any value $x$ from a normal distribution into a $z$-score for the standard normal distribution.
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What does the standard normal distribution table (in MF19/MF20) typically provide?
The cumulative probability from the far left up to a positive value $z$. This is written as $P(Z < z)$ or $\Phi(z)$.
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How do you find $P(Z > a)$ for a positive value $a$ using the standard tables?
Use the fact that the total area is 1: $P(Z > a) = 1 - P(Z < a) = 1 - \Phi(a)$.
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How do you find $P(Z < -a)$ for a positive value $a$?
By symmetry, the area in the left tail below $-a$ is equal to the area in the right tail above $+a$. So, $P(Z < -a) = P(Z > a) = 1 - \Phi(a)$.
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What are the conditions for approximating a binomial distribution $B(n, p)$ with a normal distribution?
Both $np > 5$ and $n(1-p) > 5$. This ensures that the binomial distribution is reasonably symmetric and not too skewed.
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What is a continuity correction and why is it needed?
It's an adjustment made when approximating a discrete distribution (like binomial) with a continuous one (normal). For example, the discrete probability $P(X=10)$ is approximated by the continuous area $P(9.5 < Y < 10.5)$.
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When approximating $B(n,p)$ with $N(\mu, \sigma^2)$, what are the values of $\mu$ and $\sigma^2$?
The mean is $\mu = np$ and the variance is $\sigma^2 = np(1-p)$, which are the mean and variance of the original binomial distribution.
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What is an 'inverse normal' problem?
A problem where you are given a probability (or area under the curve) and asked to find the corresponding value of the variable, $x$. This involves working backwards from the probability tables.
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What is a common mistake when using the normal distribution?
Forgetting that the parameter is the variance, $\sigma^2$. If you are given the standard deviation, $\sigma$, you must square it for the notation $N(\mu, \sigma^2)$. Conversely, if given the variance, you must take the square root for the standardisation formula.