Problem 6 uniform probability How do we assign probabilities

Problem 6 [uniform probability] How do we assign probabilities in a random experiment? From a strictly mathematical point of view, probabilities can be assigned in any way that allows the \'\'probability measure,\'\' P, to adhere to the axioms of probability (see lecture notes). When an outcome is said to occur \'\'at random\'\' (or, for emphasis, something like \'\'completely at random\'\' ), this ordinarily means that all outcomes have equal probability. This is what we call a uniform probability distribution and it corresponds to the case in which there is complete lack of knowledge about how the experiment will turn out. Let\'s consider such an experiment. Consider a tire on an automobile which is labeled around its perimeter by all angles away from an arbitrary point called 0 radians. If we traverse around the perimeter, starting at 0, moving clockwise, we will encounter labels 0,1pi,0.2pi,0.3pi,... every pi/10 radians. This labeling continues until the point labeled 1.9 pi, and, of course, at the next labeled point, we are back to 0 (2pi ) radians. Even though there are only 20 labels on the tire, there are, of course, an infinite number of points around the perimeter, one for each value of theta in the range 0

Solution

The values taken by theta are infinitely many and can be from [0, 2pi). Thus, it is a coninuous and uniform distribution.

a) Sample space S = [0,2pi) all real values in that interval.

b) The values theta will take are all the real values in the interval [0, 2pi). The probability of theta being in the given range will b given proportionate to the length of the interval. It will be ratio of the length of the interval to the length of the sample space.

c) Given a uniform distribution, P( 0.4pi < theta < 0.7pi)

= (0.7pi - 0.4pi) / (2pi) which is the normal definitiopn of probability. Desired outcome/ sample space.

= 0.3pi / 2pi

= 0.3/2

= 0.15 is the required probability.

d) Probability of a distinct point in a contiuous distribution = 0

Hence, P(theta = 0.58pi) = 0

Hope this helps. Ask if you have any doubts.