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This question on Cross Validated provides an excellent illustration for what I am going to ask. Could you please explain to me why the probability density function of a sine wave looks like it does, i.e. like a basket with the greatest probability density located at -1 and 1 points, decreasing towards the centre? Since each value appears twice over one period of the sine wave, wouldn't the probability of each value be the same? The wave is sampled along y-axis.

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avg
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  • It all depends on what you mean by the "probability density of a sine wave." If you have in mind sampling the graph evenly across its length, you will get one answer; but if you sample it evenly across the horizontal axis, you will get another. Could you tell us more specifically about how you intend to sample points on a sine wave? – whuber Feb 29 '16 at 21:26
  • I assume he means $y$-axis judging be the reference to another question and the answer he gave – Aksakal Feb 29 '16 at 21:30
  • @whuber Aksakal has an answer that I think fits my question. The context of the problem is sampling a sine wave and comparing it against ideal probability density function. Thank you for the response, too! – avg Feb 29 '16 at 21:48

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The short answer is because the sine wave is the slowest at the peak. If you imagine a pendulum (which is basically a sine wave), it's slowest at the extremes, and fastest at the bottom. As a sine wave the pendulum's bottom is the zero, and the extremes are -1 and 1.

Since, the pendulum is slowest at the extremes it spends more time at the extremes, hence you're more likely to find it there

Aksakal
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  • Thank you for the clear and simple explanation! It's interesting to see how physical properties are applicable to signal analysis. – avg Feb 29 '16 at 21:53
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    @agrus, you're welcome. Most signals come from physical processes, so no wonder. – Aksakal Feb 29 '16 at 22:00
  • Exactly! I would've never thought of a pendulum analogy. Thinking of a static picture of a sine wave makes it tricky to understand the probability density characteristics. – avg Feb 29 '16 at 22:05
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    @agrus, the static analogy would go like this: take a wire and shape it as half period of sine way. Then cut it into N such pieces that each piece would correspond to $1/N$ of the $y$-axis. You'll notice that pieces corresponding to -1 and 1 are the heaviest, i.e. the density per $y$-axis unit is the highest. – Aksakal Feb 29 '16 at 22:12
  • I'm a little confused again :) Hopefully the analogy doesn't change if I draw it on paper. Given a half cycle sine wave, if we divide the waveform into N parts, then we would end up with almost horizontal segments around the value of `1` and with segments at an angle equal to the derivative of the sine wave as we move up or down, which spawn the same horizontal distance as the straight segments. Therefore, they are going to be longer and denser because they also cover some vertical distance. Could you please let me know where I made a mistake in my assumptions? – avg Feb 29 '16 at 22:36
  • All it takes to spot a mistake is to ask a question. I totally misinterpreted it - confusing x and y axes all along. Now it's clear. Thank you once again! – avg Feb 29 '16 at 22:41
  • @agrus, the confusion is that horizontally the segments will be shorter around zero,while vertically the same, hence the total length is shorter around zero. – Aksakal Feb 29 '16 at 22:43