Question

3. Suppose that a random variable X has the density function f (x) = 2x, 0...

3. Suppose that a random variable X has the density function
f (x) = 2x, 0 ≤ x ≤ 1,
and that f (x) = 0 for x <0 and x > 1.
a) Calculate the distribution function for X, as well as the corresponding E (X) and variance V (X).

b) The numbers ​​u1 = 0.0503, u2 = 0.9149, u3 = 0.3103, u4 = 0.1866, u5 = 0.6553
uniformly distributed, independent random numbers on the interval [0, 1]. Calculate, with justifications, five independent random numbers x1,. . . , x5 for the variable X. 
c) Calculate sample mean and sample variance for the simulated numbers for
X. Compare with the result in (a). (2p)

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