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For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable X, (ii) find the cdf, F(x) = P(X ≤ x), (iii) sketch graphs of the pdf f(x) and the cdf F(x), and (iv) find μ and σ2: (a) f(x) = 4xc, 0 ≤ x ≤ 1. (b) f(x) = c √x, 0 ≤ x ≤ 4. (c) f(x) = c/x3/4, 0 < x < 1

Respuesta :

Answer:

(a)

c=1/2

cdf  is    [tex]F(x) = x^2[/tex]

Step-by-step explanation:

Remember that if   [tex]f[/tex]  is the pdf of a random variable X .

[tex]\int\limits_{-\infty}^{\infty} f(x) \,dx = 1[/tex]

Then,

(a)

[tex]\int\limits_{-\infty}^{\infty} 4xc \,dx = \int\limits_{0}^{1} 4xc \, d x = 2c = 1[/tex]

Therefore

c=1/2

and

 [tex]f(x)=2x[/tex]

then to compute the cdf, we have

[tex]F(x) = P(X \leq x) = \int\limits_{-\infty}^{x} 2y \,dy = \int\limits_{0}^{x} 2y \, dy = x^2[/tex]

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