For a sequence of real numbers x1, x2, ... ,xn, suppose random variable X takes on thesen values with equal probability 1/n. Then the first and second moment of X are defined as follows: EX = - n. EX? x? The Variance of X is defined as the average value of the squared deviations from the mean: Var(X) -> (x; - EX)? i=1 Expand the square (x; – EX)² and manipulate the summations to arrive at the following formula: Var(X) = EX² – (EX)² Show also that > (x; – EX)² = x;² – n(EX)² (=1 i=1
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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