Question

10. Determine the probability density function for the cumulative distribution function

shown below.

F {0, x < −2

0.25x + 0.5, −2 ≤ x < 1

0.5x + 0.25, 1 ≤ x < 1.5

1 x > 1.5

Determine also the following probabilities.

(a) P(X > −1)

(b)P(X < 1.3)

(c) P(−1.5 ≤ X ≤ 1.8)

(d)P(X < 1.5)

Answer 1

f(x) - density function, then f(x)=F'(x), so

$f(x)=(0.25x+0.5)'=0.25, -2≤x<1$

$f(x)=(0.5x+0.25)'=0.5, 1≤x<1.5$

$f(x)=0,$ otherwise

a) $P(X>-1)=1-P(X≤-1)=1-F(-1)=1-0.25*(-1)-0.5=0.75$

b) $P(X<1.3)=F(1.3)=0.5*1.3+0.25=0.9$

c) $P(-1.5≤X≤1.8)=F(1.8)-F(-1.5)=1-0.25*(-1.5)-0.5=0.875$

d) $P(x<1.5)=F(1.5)=1$