I need help statistic tnx.. ?

Question

1.) the waiting time, in hours, between sucessive speeders spotted by a radar unit is a continuos random variable with cumulative distribution function

F(x)={0 x<0,

1-e^-8x, x>0.

find the probability of waitng less than 12 minutes between sucessive speeders

a. using the cumulative distribution function of X;

b. using the probability density function of X.

2.An insurance company affers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X be the number of months between sucessive payments. The cumulative distribution function of X is

F(x)= {0, if x<1,

0.4, if 1<=x<3,

0.6, if3<=x<5,

0.8, if5<=x<7,

1.0, ifx>=7.

a. What is the probability mass function of X?

b. Compute P(4<X<=7)

Answer 1

1)

a)12 minutes = 12/60 hours = 0.2 hours

F(12) = 1-e^-8(0.2) = 1-e^-1.6 = 0.7981

b)

f(x) = d/dx ( 1-e^-8x) = -e^-8x(-1) = 8e^-8x , x&gt;0

P( x &lt; 0.2) =8 ∫ e^-8x , limits { 0,0.2}

= 8 e^-8x  / 8 = -e^-8x , limits  {0, 0.2}

= -e^-8(0.2) +1

=1-e^-1.6 = 0.7981

2) The probability mass function is

f(x) =0 , x &lt; 1

=0.4 , 1 &lt; x &lt; 3

=0.2 , 3 &lt; x &lt; 5

=0.2 , 5 &lt; x &lt; 7

= 0.2 , x &gt; 7

P( 4 &lt; x &lt; 7) = F(7) - F(4) = 0.8 - 0.6 = 0.2