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**1. Homework Statement**

The continuous random variable X has a uniform (rectangular) distribution on 0 ≤ x ≤ 1. Find the cumulative distributive function of X.

Three independent observations are made of X and the smallest of the three values is denoted by S.

i) Show that P(S ≤ x) = 3x - 3x² + x³ for 0 ≤ x ≤ 1

ii) Write down (or find) the value of E(S), and show that Var(S) = 3/80

The median of the three values is denoted by M. Find the probability density function of M.

**2. Homework Equations**

F(x) = ∫ f(x) dx

f(x) = 1 / (b-a) for a < x < b

Uniform distribution - E(X) = 1/2 (a + b)

Var (X) = 1/12 (b-a)²

**3. The Attempt at a Solution**

The first part is easy.

since f(x) = 1, F(X) = x for 0 ≤ x ≤ 1

F(X) =

{ 0, x < 0

{ x, 0 ≤ x ≤ 1

{ 1, x > 1

It is the part (i) and (ii) I'm stuck with.

Firstly, they said 3 independent observations and just the smallest value, which isn't known. I still don't understand why independent observations come into play here. Are this independent observations any number in the series or something? How do we know S from the other two values and determine M?

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