Seventy independent messages are sent from an electronic transmission center. Messages are processed sequentially, one after another. Transmission time of each message is Exponential with parameter λ = 5 min−1. Find the probability that all 70 messages are transmitted in less than 12 minutes. Use the Central Limit Theorem.

Following are the solution to the given question:

Let X signify the letter's transmission time   , and this is assumed that the  .

               mean  

               Variance  

       Therefore

Therefore, the final answer is "0.1170".

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The probability that all 70 messages are transmitted in less than 12 minutes is 0.117.

Step-by-step explanation:

Let X = the transmission time of each message.

The random variable X follows an Exponential distribution with parameter λ = 5 minutes.

The expected value of X is:

The variance of X is:

Now define a random variable T as:

T = X₁ + X₂ + ... + X₇₀

According to a Central limit theorem if a large sample (n > 30) is selected from a population with mean μ and variance σ² then the sum of random variables X follows a Normal distribution with mean, and variance, .

Compute the probability of T < 12 as follows:

*Use a z-table for the probability.

Thus, the probability that all 70 messages are transmitted in less than 12 minutes is 0.117.

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