Answer to Question #190645 in Statistics and Probability for Anees Rehman

Question

Binary data are transmitted over a noisy communication channel in blocks of 16 binary digits. The probability that a received binary digit is in error due to channel noise is 0.1. Assume that the occurrence of an error in a particular digit does not influence the probability of occurrence of error in any other digit with in the block. a. Find the average (or expected) number of error per block. b. Find the variance of the number of error per block. c. Find the probability that the number of errors per block is greater than or equal to 5.

Answer 1

a.) Let X be the random variable representing the number of errors per block . Then, X has a binomial distribution:

$P(X = k ) = ^{16}C_k(0.1)^k(0.9)^{16-k}$

$E(X) = np = 16 \times 0.1 = 1.6$

b.) Variance of X = $\sigma_X^2 = np(1-p) = 16\times 0.1 \times 0.9 = 1.44$

c.) The probability that the number of errors per block is greater than or equal to 5.

$= P(X \ge 5) = 1-P(X<5)$

$= 1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)]$

$= 1- [^{16}C_0(0.1)^0(0.9)^{16}+^{16}C_1(0.1)^1(0.9)^{15}+^{16}C_2(0.1)^2(0.9)^{14}+^{16}C_3(0.1)^3(0.9)^{13}+^{16}C_0(0.1)^4(0.9)^{12}]$