Errors in an experimental transmission channel are found when the transmission i

Question

Errors in an experimental transmission channel are found when the transmission is checked by a certifier that defects missing pulses. The number of errors found in an eight bit byte is a random variable with the following distribution: Determine the probability mass function and determine each of the following probabilities: Determine the mean and variance of X. The TOEFL requirement for admission at the University is 6 or more points. It was found that the probability that a student gets 6 or more points is 0.3. The student can repeat the test as many times as he/she wishes till they achieve the requirements.

Explanation / Answer

Here we have given CDF.

From CDF we can find pdf by subtraction.

f(x) = 0 x < 1

= 0.1 1 x < 2

= 0.6 2 x < 6

= 0.3 6 x

P(X 3.5) = 0.6 3.5 lies between 2 and 6.

P(X > 5) = 1 - P(X 5) = 1 - F(5) = 1 - 0.7 = 0.3 5 lies between 2 and 6.

P(X 5.2) = 0.6 0.6 lies between 2 and 6.

P(X >7) = 1 - P(X 7) = 1 - F(7) = 1 - 1 = 0

P(1 X 2) = P(X 2) - P(X 1) = F(2) - F(1) = 0.7 - 0.1 = 0.6

P(X < 2) = 0.1

P(X=2) = 0.6

P(X 2) = F(2) = 0.7

Mean and variance of X :

E(X) = x*p(x) = 1*0.1 + 2*0.6 + 6*0.3 = 3.1

To find variance we need E(X^2).

E(X^2) = x^2*p(x)

= 1^2 * 0.1 + 2^2*0.6 + 6^2*0.3 = 13.3

V(X) = E(X^2) - (E(X))2 = 13.3 - 3.1^2 = 3.69

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