Can someone give me real world examples of uniform distribution on [0,1] of a continuous random variable, because I could not make out one.
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$\begingroup$Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, measuring the resulting angle (divided by $2\pi$) as a number from $0$ to $1$.
$\endgroup$ 2 $\begingroup$You show up at a bus stop to wait for a bus that comes by once per hour. You do not know what time the bus came by last. The arrival time of the next bus is a continuous uniform distribution [0,1] measured in hours.
$\endgroup$ $\begingroup$In analog-to-digital conversion a quantization error occurs. This error is either due to rounding or truncation. When the original signal is much larger than one least significant bit (LSB), the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution. The RMS error therefore follows from the variance of this distribution.
$\endgroup$ $\begingroup$This is like the spinner example. Consider throwing a dart at a dart board. Assuming that all directions are equally likely, the angle of deflection from the x-axis drawn through the bullseye should be uniformly distributed between $0$ and $360^\circ$ or $0$ and $2 \pi$. Rescaling would produce a uniform $0$, $1$.
$\endgroup$ $\begingroup$Well to simulate a normal distribution in Excel one first simulates a uniform random variable $U \sim Unif(0,1)$ and then computes the value $x$ s.t.
$$U = \int_{-\infty}^{x} f_X(t) dt$$
where $f_X(t)$ is the pdf of a normal distribution with mean and std dev $(\mu, \sigma^2)$
Essentially, the simulation of a number in $(0,1)$ here is the simulation of a probability.
$\endgroup$ 2 $\begingroup$If there are N raffle tickets sold to N different people, then each ticket holder has the same probability of winning the prize. The distribution is uniform.
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