S2.3 Continuous uniform distribution and Normal approximations

What is the continuous uniform distribution?

Some quantities are equally likely to be anywhere in a fixed range and cannot fall outside it: the waiting time at a crossing whose signal cycles regularly, the point where a rod snaps, the error left after rounding a reading. The model for all of these is the simplest pdf there is — a flat line.

• The graph of \(f\) is a rectangle of width \(b-a\) and height \(\frac{1}{b-a}\), so its total area is 1, as every pdf must have.
• Probabilities are areas of thinner rectangles, so for \(a\le c\lt d\le b\): \(P(c\lt X\lt d)=\dfrac{d-c}{b-a}\). No integration is needed.
• For any continuous variable \(P(X=c)=0\), so \(\lt\) and \(\le\) give the same probability. (This is not true for the binomial and Poisson distributions later on this page.)
• The distribution is symmetrical about its midpoint, so the mean and the median are both \(\frac{a+b}{2}\). It has no mode: every value in the range is equally likely.

Where do the mean and variance of U[a, b] come from?

Both come from the general results for a continuous random variable in S2.2: \(E(X)=\int xf(x)\,dx\) and \(\text{Var}(X)=\int x^2f(x)\,dx-[E(X)]^2\). The only algebra you need is two factorisations.

How do you use the cumulative distribution function?

The cdf is \(F(x)=P(X\le x)\). For \(U[a,b]\), integrate the constant pdf from \(a\) to \(x\):

• Quartiles and percentiles: solve \(F(x)=0.25,\ 0.5,\ 0.75,\ldots\). The \(r\)th percentile is \(a+\frac{r}{100}(b-a)\), a fraction of the way along the interval.
• If a question gives the mean and variance, use them as two simultaneous equations: \(a+b=2E(X)\) and \((b-a)^2=12\,\text{Var}(X)\). Take the positive square root, since \(b\gt a\).

How does the uniform distribution model rounding errors?

When a measurement is rounded to the nearest \(k\), the error (true value − recorded value) can be anything from \(-\frac{k}{2}\) to \(\frac{k}{2}\), with no reason to favour any part of that range. So the rounding error is modelled by \(U\left[-\frac{k}{2},\frac{k}{2}\right]\):

• to the nearest whole number: \(U[-0.5,0.5]\); to the nearest 0.1: \(U[-0.05,0.05]\); to the nearest 5: \(U[-2.5,2.5]\);
• the mean error is 0 and the variance is \(\frac{k^2}{12}\);
• if a value is truncated (chopped down) rather than rounded, the error is \(U[0,k]\) instead, and its mean is \(\frac{k}{2}\), not 0.

Other natural uniform models are the position of a random break along a rod, the time of an event during a fixed interval, and an angle chosen at random.

Expected values of functions: E(g(X))

\(E(g(X))=\int g(x)f(x)\,dx\). For linear functions use \(E(cX+d)=cE(X)+d\) and \(\text{Var}(cX+d)=c^2\text{Var}(X)\). For \(E(X^2)\) the quickest route is \(E(X^2)=\text{Var}(X)+[E(X)]^2\). Example: a disc has radius \(R\) cm with \(R\sim U[2,4]\). Then \(E(R^2)=\frac{4}{12}+3^2=\frac{28}{3}\), so the expected area is \(E(\pi R^2)=\frac{28\pi}{3}\approx29.3\) cm², a little more than \(\pi\times3^2=9\pi\).

When can a Normal distribution approximate a binomial distribution?

The binomial tables stop at a limited set of values of \(n\) and \(p\), and summing dozens of binomial terms by hand is impractical. When the binomial distribution is roughly bell-shaped, a Normal curve with the same mean and variance gives an accurate shortcut. That happens when \(n\) is large and \(p\) is close to \(\frac12\). As a working guide, check that \(np\) and \(n(1-p)\) are both greater than 5.

If \(p\) is small (or close to 1), the binomial distribution is skewed and the Normal curve fits badly; with \(p\) small and \(n\) large, use the Poisson approximation from S2.1 instead.

What is a continuity correction, and how do you apply it?

A binomial or Poisson variable takes only whole-number values; a Normal variable is continuous. Think of each whole number \(k\) as a bar from \(k-0.5\) to \(k+0.5\). The continuity correction replaces "these whole numbers" by "the whole width of their bars". Decide which integers are included, then go half a unit beyond the outermost ones:

For the continuous \(Y\), \(\lt\) and \(\le\) are the same, so it does not matter which you write on the right-hand side. Without a continuity correction, \(P(X=50)\) would become \(P(Y=50)=0\), which is plainly wrong. The correction is what makes the approximation work.

When can a Normal distribution approximate a Poisson distribution, and which approximation should you choose?

The Poisson distribution has mean and variance both equal to \(\lambda\). As \(\lambda\) grows, its slight right skew fades and it becomes bell-shaped. For large \(\lambda\) (as a guide, \(\lambda\gt10\)) use \(N(\lambda,\lambda)\), with a continuity correction as before, since a Poisson variable is also a whole-number count. If the rate is given per unit of time or space, scale \(\lambda\) first (2.5 per minute → 30 per 12 minutes), then approximate.

Choosing the approximation

If \(p\) is close to 1, count failures instead: the number of failures is \(B(n,1-p)\) with a small probability. Use the exact tables whenever \(n\), \(p\) or \(\lambda\) are in them: an approximation is only for when the exact route is impractical.

How does a full exam question fit together?

The two halves of this topic often meet in one question: a uniform model gives a probability \(p\), a count of successes over many repetitions is then binomial with that \(p\), and the Normal approximation answers questions about the count — including a "find the largest value" part that runs the method backwards with a percentage point.

Exam tips for S2.3

Common misconceptions

How this topic connects

S1.6 The Normal distribution

Every approximation on this page ends with standardising and the \(\Phi(z)\) or percentage-point tables, exactly as in S1; only the continuity correction is new.

S2.1 and S2.2

The approximations replace the exact binomial and Poisson models from S2.1, and the uniform distribution is the simplest case of the general pdf, cdf, \(E(X)\) and \(\text{Var}(X)\) methods from S2.2.

S2.4 Hypothesis tests

Tests on a binomial proportion or a Poisson rate with large numbers use these same Normal approximations to find critical regions. In IAL Physics, a reading rounded to the smallest scale division has an error spread evenly over ± half a division, and large radioactive counts behave approximately Normally with standard deviation \(\sqrt{\text{count}}\).

Key formulae

Check your understanding

Specification checklist

• (S2 3.1) I can write down the pdf of a continuous uniform distribution on \([a,b]\), with its range, and find probabilities from it.
• (S2 3.1) I can derive and specify fully the cumulative distribution function of a uniform distribution and use it to find quartiles and percentiles.
• (S2 3.1) I can derive the mean \(\frac{a+b}{2}\) and variance \(\frac{(b-a)^2}{12}\) by integration, and use them to find unknown parameters.
• (S2 3.1) I can find \(E(g(X))\) for a uniform variable, including \(E(X^2)\) and linear functions.
• (S2 3.1) I can model rounding errors and other "equally likely anywhere in a range" situations with a uniform distribution.
• (S2 3.2) I can approximate \(B(n,p)\) by \(N(np,np(1-p))\) and state when this is suitable.
• (S2 3.2) I can approximate \(Po(\lambda)\) by \(N(\lambda,\lambda)\) for large \(\lambda\), scaling the rate first where needed.
• (S2 3.2) I can apply the correct continuity correction to any inequality or single value.
• (S2 3.2) I can choose the most appropriate approximation (Poisson or Normal) for a given situation and justify the choice, and use an approximation in reverse with a percentage point to find an unknown.