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Alekssandra [29.7K]
3 years ago
10

In a certain communications system, there is an average of 1 transmission error per 10 seconds. Assume that the distribution of

transmission errors is Poisson. The probability of 1 error in a period of one-half minute is approximately ________.
Mathematics
2 answers:
trasher [3.6K]3 years ago
8 0

Answer:

The probability of 1 error in a period of one-half minute is approximately 0.15 .

Step-by-step explanation:

We are given that in a certain communications system, there is an average of 1 transmission error per 10 seconds.

Let X = distribution of transmission errors

So, X ~ Poisson(\lambda) , where \lambda = average transmission error per 10 seconds = 1

i.e; X ~ Poisson(\lambda = 1)  

The Probability distribution of Poisson distribution is given by;

P(X=x) = \frac{e^{-\lambda} \times \lambda^{x}  }{x!} ; x = 0,1,2,3,....

Since we have to find the probability for a period of one-half minute and we are given for a period of per 10 seconds.

Firstly, we need to convert \lambda into period of one-half minute(30 seconds), i.e;

             \lambda for per 10 seconds period = 1

             \lambda for 1 second period = \frac{1}{10}

             \lambda for 30 second period = \frac{1}{10} \times 30 = 3 errors

So, required X ~ Poisson(\lambda=3)

Now, probability of 1 error in a period of one-half minute = P(X = 1)

  P(X = 1) = \frac{e^{-3} \times 3^{1}  }{1!} = 3 \times e^{-3} = 0.1494

Therefore, probability of 1 error in a period of one-half minute is approximately 0.15 or 15% .

Nitella [24]3 years ago
7 0

Answer:

The probability of 1 error in a period of 0ne - half minute is 0.1494

Step-by-step explanation:

Formula for poisson distribution:

P (X = k) = \frac{\exp^{- \lambda} \lambda^{x}  }{x!}

If there is an average of 1 error in 10 seconds

In one-half minutes (i.e. 30 seconds), there will be an average of 30/10 errors = 3 errors

\lambda = 3 errors\\x = 1

P (X = 1) = \frac{\exp^{-3} 3^{1}  }{1!}

1! = 1

P (X = 1) = 3 \exp^{-3}

P(X = 1) = 3 * 0.0498

P(X = 1) = 0.01494

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