# Chaitanya's Random Pages

## October 30, 2013

### Sparsely populated regions of Australia

Filed under: geography — ckrao @ 1:11 pm

Upon seeing a population density map of Australia at the Australian Bureau of Statistics site here, I wanted to find out how much of the country has a populaton density less than 0.1 person per square kilometre. This corresponds to the white region below (map uses shapefiles from here and 2012 population estimation data from here). Several of the red regions in the outback only just missed the cutoff having slightly more than 0.1 people per square kilometre.

Red regions have population density > 0.1/sq km

The table below gives population details of the numbered regions in the map above. We see that the sparsest region is western South Australia (3 – Western) with just 112 people in an area close to 75% of the entire state of Victoria! Regions labelled 7,8 of WA, 2,3 of SA, 5,6 of NT, 3,7,9 of QLD and 1 of NSW are the regions with less than 0.02 people per square kilometre.

In summary almost 5.6 million square kilometres, or 73% of the land area of Australia, is occupied by just over 140 thousand people, or 0.6% of the population! The unlabelled white regions are mostly mountainous regions plus the South West Wilderness of Tasmania.

Numbered regions in more detail

 State Number on map Area Name Population Area (sq km) Density (people/sq km) WA 1 Kununurra 8,490 117,663.8 0.0722 2 Roebuck 2,489 55,602.9 0.0448 3 Derby – West Kimberley 9,258 110,883.6 0.0835 4 Halls Creek 3,968 135,357.6 0.0293 5 East Pilbara 8,000 389,540.9 0.0205 6 Exmouth 4,197 134,986.0 0.0311 7 Meekatharra 4,263 414,506.0 0.0103 8 Leinster – Leonora 6,050 496,740.6 0.0122 9 Mukinbudin 3,554 50,125.2 0.0709 10 Kambalda – Coolgardie – Norseman 5,739 217,989.9 0.0263 11 Esperance Region 4,430 55,408.3 0.0800 NT 1 Daly 2,270 34,793.8 0.0652 2 Elsey 2,380 92,949.5 0.0256 3 Gulf 4,741 92,351.7 0.0513 4 Victoria River 2,864 133,608.5 0.0214 5 Barkly 3,089 303,252.5 0.0102 6 Tanami 3,423 192,631.2 0.0178 7 Yuendumu – Anmatjere 2,392 71,841.5 0.0333 8 Sandover – Plenty 4,354 129,514.9 0.0336 9 Petermann – Simpson 2,497 175,251.0 0.0142 SA 1 APY Lands 2,692 102,331.5 0.0263 2 Outback 3,516 519,520.0 0.0068 3 Western 112 168,690.3 0.0007 QLD 1 Cape York 7,602 113,022.7 0.0673 2 Carpentaria 5,340 114,279.0 0.0467 3 Croydon – Etheridge 1,249 68,688.1 0.0182 4 Mount Isa Region 4,097 84,072.8 0.0487 5 Northern Highlands 3,761 108,506.6 0.0347 6 Dalrymple 3,991 68,331.8 0.0584 7 Far Central West 2,515 271,262.1 0.0093 8 Barcaldine – Blackall 5,602 83,909.7 0.0668 9 Far South West 3,342 188,802.5 0.0177 NSW 1 Far West 2,820 146,691.0 0.0192 2 Bourke – Brewarrina 4,524 56,850.2 0.0796 3 Wentworth-Balranald Region 3,742 49,724.6 0.0753 Totals from labelled white region 143,353 5,549,682.3 0.0258 Totals from unlabelled white region 289 41,521.7 0.0070 Totals from white region 143,642 5,591,204.0 0.0257

The above numbered regions do not include the following outback towns which are listed below (seen mostly as dots in the above map). As can be seen their total population is comparable to that of the vast white region shown above. Note that towns such as Cobar, Newman, Cloncurry and Kununurra are larger than Coober Pedy but still are part of the white numbered regions.

 Town Population Area (sq km) Density Kalgoorlie-Boulder 32,859 102.7 320.0 Alice Springs 28,605 327.6 87.3 Mount Isa 21,945 62.8 349.4 Broken Hill 19,103 170.3 112.2 Charters Towers 8,440 41.7 202.4 Roxby Downs 4,953 281.2 17.6 Tennant Creek 3,570 42.1 84.8 Coober Pedy 1,768 77.7 22.8 Totals 121,243 1,106.1 109.6

Even after including these towns, we have just over a quarter of a million people (1.1% of the nation’s population) in such a vast connected region.

## October 24, 2013

### A lower bound for the tail probability of a normal distribution

Filed under: mathematics — ckrao @ 11:17 am

If the random variable $X$ has a normal distribution with mean 0 and variance 1, we know that its tail probability $\text{Pr}(X >x)$ is given by the integral

$\displaystyle \text{Pr}(X >x) = \frac{1}{\sqrt{2\pi}} \int_x^{\infty} \exp (-t^2/2)\ \text{d}t$

(that is, the area under a standard normal bell-shaped curve from $x$ up to $\infty$).

This integral does not have a closed form in terms of elementary functions (unless $x = 0$). However we can find a good lower bound as

$\displaystyle \frac{\sqrt{4 + x^2} - x}{2} \cdot \frac{1}{\sqrt{2\pi}} \exp (-x^2/2) \leq \text{Pr}(X >x) \quad x > 0. \quad \quad (1)$

We show the proof of this result based on [1]. Interestingly it involves Jensen’s inequality: recall that this states that if $f$ is a convex function and ${\mathbb E}$ denotes expectation with respect to some probability distribution, then $f({\mathbb E} Y) \leq {\mathbb E} f(Y)$ for any random variable $Y$. Applied to the convex function $f(u) = 1/u (u > 0)$ this becomes

$\displaystyle \frac{1}{{\mathbb E} Y} \leq {\mathbb E} \frac{1}{Y}. \quad \quad (2)$

For fixed $x > 0$ we now let $Y$ have the distribution

$\displaystyle f_Y(t) = \begin{cases}\frac{te^{-t^2/2}}{\int_x^{\infty} te^{-t^2/2}\ d\text{t} } & \text{if } t \geq x,\\ 0 & \text{if } t < x. \end{cases} \quad \quad (3)$

Applying this in (2) gives

$\displaystyle \frac{\int_x^{\infty} te^{-t^2/2}\ d\text{t}}{\int_x^{\infty} t^2 e^{-t^2/2} \ d\text{t}} \leq \frac{\int_x^{\infty} e^{-t^2/2} \ d\text{t}}{\int_x^{\infty} te^{-t^2/2}\ d\text{t} } \quad \quad (4)$

Evaluating these terms,

$\displaystyle \int_x^{\infty} te^{-t^2/2}\ d\text{t} = \left[-e^{-t^2/2} \right]_x^{\infty} = e^{-x^2/2}$

and

\begin{aligned} \int_x^{\infty} t^2 e^{-t^2/2}\ d\text{t} &= \left[-t e^{-t^2/2} \right]_x^{\infty} - \int_x^{\infty} e^{-t^2/2}\ d \text{t}\\ &= xe^{-x^2/2} + \int_x^{\infty} e^{-t^2/2}\ d \text{t},\end{aligned}

so (4) becomes

$\displaystyle \left(e^{-x^2/2}\right)^2 \leq \left(xe^{-x^2/2} + \int_x^{\infty} e^{-t^2/2}\ d \text{t}\right) \left(\int_x^{\infty} e^{-t^2/2} \ d\text{t}\right).$

This is a quadratic inequality in $a(x):= \left(\int_x^{\infty} e^{-t^2/2} \ d\text{t}\right)$ being of the form

$\displaystyle c(x) \leq (b(x) + a(x))a(x),$

where $c(x) = \left(e^{-x^2/2}\right)^2$ and $b(x) = xe^{-x^2/2}$.

This is equivalent to $c(x) + b^2(x)/4 \leq (a(x) + b(x)/2)^2$, or

$\displaystyle \sqrt{c(x) + b^2(x)/4} - b(x)/2 \leq a(x)$

since the quantities involved are non-negative. In other words we have

$\displaystyle e^{-x^2/2} \frac{\sqrt{4 + x^2} - x}{2} \leq \int_x^{\infty} e^{-t^2/2} \ d\text{t},$

which is equivalent to (1), as desired.

Another lower and upper bounds for the tail probability of a normal distribution are

$\displaystyle \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} \exp(-x^2/2) \leq \text{Pr}(X >x) \leq \frac{1}{x}\frac{1}{\sqrt{2\pi}} \exp(-x^2/2),$

a proof of which can be seen, e.g. in [2].
The bound we have looked at can be combined with the following upper bound to arrive at the following tighter bounds.

$\displaystyle \frac{1}{x+ \sqrt{4 + x^2} } \leq \sqrt{\frac{\pi}{2}}\exp (x^2/2) \text{Pr}(X >x) \leq \frac{1}{\sqrt{x+ 8/\pi + x^2}}$

See [3] and [4] for a derivation of the upper bound as well as similar bounds.

#### References

[1] Z. W. Birnbaum, An Inequality for Mill’s Ratio, Ann. Math. Statist. Volume 13, Number 2 (1942), 245-246.

[2] J. D. Cook, Upper and lower bounds for the normal distribution function, 2009.

[4] L. Duembgen, Bounding Standard Gaussian Tail Probabilities, University of Bern Technical Report 76, 2010

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