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

Answer:

a) $P(X \leq 4)$

And we can find this using the cumulative distribution function:

$P(X \leq 4) = F(4) = 0.9$

b) $P(X > 7)$

And we can find this using the cumulative distribution function and the complement rule on this way:

$P(X >7) =1-P(X\leq 7)= 1- F(7) = 1-1 = 0$

c) $P(X \leq 5)$

And we can find this using the cumulative distribution function:

$P(X \leq 5) = F(5) = 0.9$

d) $P(X > 4)$

And we can find this using the cumulative distribution function and the complement rule on this way:

$P(X >4) =1-P(X\leq 4)= 1- F(4) = 1-0.9 = 0.1$

e) $P(X \leq 2)$

And we can find this using the cumulative distribution function:

$P(X \leq 2) = F(2) = 0.7$

Step-by-step explanation:

For this case we have the following cumulative distribution function:

$F(x) = 0 , x$

$F(x) = 0.7, 1 \leq x$

$F(x) = 0.9, 4 \leq x$

$F(x) = 1, x \geq 7$

Part a

We want this probability:

$P(X \leq 4)$

And we can find this using the cumulative distribution function:

$P(X \leq 4) = F(4) = 0.9$

Part b

We want this probability:

$P(X > 7)$

And we can find this using the cumulative distribution function and the complement rule on this way:

$P(X >7) =1-P(X\leq 7)= 1- F(7) = 1-1 = 0$

Part c

We want this probability:

$P(X \leq 5)$

And we can find this using the cumulative distribution function:

$P(X \leq 5) = F(5) = 0.9$

Part d

We want this probability:

$P(X > 4)$

And we can find this using the cumulative distribution function and the complement rule on this way:

$P(X >4) =1-P(X\leq 4)= 1- F(4) = 1-0.9 = 0.1$

Part e

We want this probability:

$P(X \leq 2)$

And we can find this using the cumulative distribution function:

$P(X \leq 2) = F(2) = 0.7$