Question

3. Each time a modem transmits one bit, the receiving modem analyzes the signal arrives and decides whether the transmitted bit is 0 or 1. It makes an error with probability p, independent of whether any other bit is received correctly.

a) If the transmission continues until first error, what is the distribution of random variable X, the number of bits transmitted?

b) If p = 0.1, what is probability that X = 10?

c) what is probability that X > 5?

d) If modem transmits 30bits, what is the distribution of random variable Y, the number of error?

e) If p = 0.15, what is probability that Y< 3,

f) if the transmission continues until third error ,what is the distribution of random variable Z, the number of bits transmitted?

g) If p = 0.2, what is probability that Z = 5?

h) what is probability that Z > 5?

Answer 1

a)

The distribution of random variable X is Geometric distribution with parameter p.

b)
X ~ Geometric(p = 0.1)

for k = 1, 2, 3..

c)

d)

The the distribution of random variable Y is Binomial distribution with parameter n = 30 and probability as p.

e)

Y ~ Binomial(n = 30, p = 0.15)

P(Y< 3) = P(Y = 0) + P(Y = 1) + P(Y = 2)

= 0.00763076 + 0.04039814 + 0.10337171

= 0.1514

f)

The distribution of random variable Z is negative binomial with parameters r = 3 and probability as p.

g)

Z ~ NegBinomial(r = 3, p = 0.2)

for z = 3, 4, 5,..

h)

P(Z > 5) = 1 - P(Z 5) = 1 - (P(Z= 3) + P(Z = 4) + P(Z = 5))

= 1 - 0.05792

= 0.94208