Let x be a random variable defined by the density function f

Let x be a random variable defined by the density function f(x) Find E(X) Find the variance and standard deviation for the random variable x.

Solution

a)

Here,

E(x) = Integral [x f(x)]

= Integral [3x^3 dx] | (0,1)

= 3x^4 / 4 |(0,1)

= 3/4 [answer]

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b)

Also,

E(x^2) = Integral [x^2 f(x)]

= Integral [3x^4 dx] | (0,1)

= 3x^5 / 5 |(0,1)

= 3/5

Thus, as

var(x) = E(x^2) - E(x)^2

then

var(X) = 3/5 - (3/4)^2 = 0.0375 [answer, variance]

Also,

s(x) = sqrt(var(x)) = 0.193649167 [answer, standard deviation]