An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows:

F(x) =

0 x < 1
0.33 1 < x < 3
0.44 3 < x < 4
0.48 4 < x < 6
0.86 6 < x < 12
1 12 < x
(a) What is the pmf of X?
x 1 3 4 6 12
p(x)

(b) Using just the cdf, compute P(3 = X = 6) and P(4 = X).
P(3 < X < 6) =
P(4 <

X) =

a)

The pmf of x is

x     p(x)

1      0.33

3     0.11

4     0.04

6     0.38

12    0.14

b)

P(3≤X≤6)=0.53

P(X≥4)=0.56

Step-by-step explanation:

a)

We have to find pmf of x.

We know that

F(x)=P(X≤ x)

F(0)=P(X≤ 0)=P(X=0)=f(0)

F(1)=P(X≤ 1)=P(X=0)+P(X=1)=f(0)+f(1)

As, f(0)=F(0), so

f(1)=F(1)-F(0)

F(2)=P(X≤ 2)=P(X=0)+P(X=1)+P(X=2)=F(1)+f(2)

f(2)=F(2)-F(1)

So, we can say that

f(3)=F(3)-F(2)

f(4)=F(4)-F(3)

f(6)=F(6)-F(5)

f(12)=F(12)-F(11).

We are given that

F(0)=0 ,F(1)=0.33 ,F(2)=0.33 ,F(3)=0.44, F(4)=0.48, F(5)=0.48, F(6)=0.86, F(7)=0.86, F(8)=0.86, F(9)=0.86, F(10)=0.86, F(11)=0.86, F(12)=1.

We have to find f(1), f(3), f(4), f(6) and f(12).

f(1)=F(1)-F(0)=0.33-0=0.33

f(3)=F(3)-F(2)=0.44-0.33=0.11

f(4)=F(4)-F(3)=0.48-0.44=0.04

f(6)=F(6)-F(5)=0.86-0.48=0.38

f(12)=F(12)-F(11)=1-0.86=0.14

The pmf of x is

x     p(x)

1      0.33

3     0.11

4     0.04

6     0.38

12    0.14

b)

P(3≤X≤6)=?

We know that P(a<X≤b)=F(b)-F(a)

P(3≤X≤6)=P(2<X≤6)=F(6)-F(2)=0.86-0.33=0.53

P(3≤X≤6)=0.53

P(X≥4)=?

P(X≥4)=1-P(X<4)=1-P(X≤3)=1-F(3)=1-0.44=0.56

P(X≥4)=0.56

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all