Question

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.

Answer 1

Answer:

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:

$E(X)=\frac{1}{\lambda}=\frac{1}{5}=0.20$

The variance of X is:

$V(X)=\frac{1}{\lambda^{2}}=\frac{1}{5^{2}}=0.04$

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, $\mu_{s} = n\mu$ and variance, $\sigma^{2}_{s}=n\sigma^{2}$.

Compute the probability of T < 12 as follows:

$P(T$

*Use a z-table for the probability.

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