Here the text...

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*suppose X_1, X_2, ... , X_100 are independent random variables with common mean "mu" and variance "sigma squared." Let X be their average. What is the probability that |X - "mu" | is greater than or equal to 0.25?*...

Very well!... in general if n is the number of random variables, the 'average' is by definition...

(1)

... i.e. their sum devided by n. Now we suppose that the

are all uniformely distributed between 0 and 1 [so that we examine a particular case, not the general case...] so that is

and

. If...

(2)

... then ...

(3)

... so that , neglecting the term n in (1) and indicating with

the p.d.f of the X, is...

(4)

... and performing the Inverse Laplace Transform of (4) we derive...

(5)

... where...

(6)

Of course for 'large n' the computation of (5) requires a computer and something like fifteen years ago I composed a specific computer program. Using this program we found that for n=100 is...

(7)

Kind regards