Binary data are transmitted over a noisy communication channel. The probability that a received binary digit is in error due to channel noise is 0.05. Assume that such errors occur independently within the bit stream. (a) What is the probability that the 3rd error occurs on the 50th transmitted bit?
(b) On average, how many bits will be transmitted correctly before the first error?
(c) Consider a 32 bit transmissions over a noisy communication channel, What is the probability of exactly 2 errors in this word?

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Step-by-step explanation:

a.) P(x=3)=0.0166

When a random variable is defined as the kth success in the nth trial, the negative binomial distribution follows.

The probability mass function for the negative binomial distribution is defined as,

P(X=k)=\binom{n-1}{k-1}p^k(1-p)^{n-k}

where

k is the kth success = 3

n is the number of trials needed for kth success = 50

p = probability of success = 0.05

P(X=3)=\binom{50-1}{32-1}0.05^3(1-0.05)^{50-3}

P(X=3)=0.0166

b) On average, 20 bits will be transmitted before the first error.

When a random variable is defined as the number of trials needed until the first success occurs, the geometric distribution follows,

The expected value of the geometric random variable is obtained using the following formula,

E[X]=\frac{1}{p}=\frac{1}{0.05}=20

c) P(X = 2) = 0.0166

The probability mass function for the negative binomial distribution is defined as,

P(X=k)=\binom{n-1}{k-1}p^k(1-p)^{n-k}

Given

k = 2

n = 32

P(X=2)=\binom{32-1}{2-1}0.05^2(1-0.05)^{32-2}

P(X = 2) = 0.0166

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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