Question

From the probability mass function f(x) =2x+1/25

for x = 0,1,2,3,4, find the

cumulative distribution function of the random variable X. Using F(x), determine

the following probabilities.

(a) P(X ≤ 1)

(b)P(2 ≤ X < 4)

(c) P(X < 3)

Answer 1

$F(x)=0$ , for $x<0$

$F(x)={\frac {2*0+1} {25}}={\frac 1 {25}}$ , for $0≤x<1$

$F(x)={\frac 1 {25}}+{\frac {2*1+1} {25}}={\frac 4 {25}}$ , for $1≤x<2$

$F(x)={\frac 4 {25}}+{\frac {2*2+1} {25}}={\frac 9 {25}}$ , for $2≤x<3$

$F(x)={\frac 9 {25}}+{\frac {2*3+1} {25}}={\frac {16} {25}}$ , for $3≤x<4$

$F(x)={\frac {16} {25}}+{\frac {2*4+1} {25}}={\frac {25} {25}}=1$ , for $x≥4$

a) $P(X≤1)=F(1)={\frac 4 {25}}$

b) $P(2≤X<4)=F(3)-F(1)={\frac {16} {25}} - {\frac 4 {25}}={\frac {12} {25}}$

c) $P(X<3)=F(2)={\frac 9 {25}}$