Question

Errors in an experimental transmission channel are found when thetransmission is checked by a certifier that detects missing pulses.The number of errors found in a eight-bit byte is a random variablewith the following distribution:

F(x) = { 0 x<1
0.7 1 <= x < 4
0.9 4 <= x < 7
1 7 <= x

Determine each of the following probabilities:
(a) P(X <= 4) (b) P(X > 7)
(c) P(X <= 5) (d) P(X > 4)
(e) P(X <= 2)

Answer 1

a) P(X ≤ 4) = 0.9

b) P(X>7) = 0

c) P(X ≤ 5) = 0.9

d) P(X>4) = 0.1

e) P(X≤2)= 0.7

Explanation:

Hello!

Given the distribution of cumulative probability

0 for X < 1

F(x) 0.7 for 1 ≤ X < 4

0.9 for 4 ≤ X < 7

1 for 7 ≤ X

To determine each of the asked probabilities you have to select the probability corresponding to the defined intervals for X:

a) P(X ≤ 4) = 0.9

This value of X is included the interval "4 ≤ X < 7" so the corresponding probability is 0.9

b) P(X>7) = 0

This is equal to the expression:

1 - P(X≤7)

This expression is included in the last interval so the probability is 1

1 - 1 = 0

c) P(X ≤ 5) = 0.9

5 is included in the third interval "4 ≤ X < 7" so the corresponding probability is 0.9

d) P(X>4) = 0.1

P(X>4) = 1 - P(X≤4)

P(X≤4) is in the interval of definition "4 ≤ X < 7" so the corresponding probability is 0.9

1 - 0.9 = 0.1

e) P(X≤2)= 0.7

The value X=2 is included in the second interval "for 1 ≤ X < 4", so the probability is 0.7

I hope it helps!