Question

The probability density function of the continuous random variable X is given by

fX (x) = kx, (0 <= x <2)

= k (4-x), (2 <= x <4)

= 0, (otherwise)

1) Find the value of k

2)Find the mean of m

3)Find the Dispersion σ²

4)Find the value of Cumulative distribution function FX(x)

Answer 1

The probability density function of a continuous random variable X is given by

fX(x) =kx   0 x < 2

=k(4-x) 2 x < 4

= 0 o.w.

1. We know that

thus

2.

The mean is given by

thus,

, putting the value of k=1/4, we have,

The mean is E(X)=2=m

3. The variance is given by,

V(X) = E(X2) - E2(X)

now, since,

V(X) = E(X2) - E2(X)

i.e.

4. The CDF is given by

again,

Thus, the required cumulative distribution function is given by