Question

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

Answer 1

Answer:

(a)

c=1/2

cdf  is    $F(x) = x^2$

Step-by-step explanation:

Remember that if   $f$  is the pdf of a random variable X .

$\int\limits_{-\infty}^{\infty} f(x) \,dx = 1$

Then,

(a)

$\int\limits_{-\infty}^{\infty} 4xc \,dx = \int\limits_{0}^{1} 4xc \, d x = 2c = 1$

Therefore

c=1/2

and

 $f(x)=2x$

then to compute the cdf, we have

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