Mathematics, 16.03.2020 19:37 ayoismeisalex

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.

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Answer from: leriscepowell3654

Following are the solution to the given question:

Let X signify the letter's transmission time $i^{th}$ , and i =1,2,3,...,70. this is assumed that the $X_i \sim Exp (\lambda =5)$ .

$\to Mean =\frac{1}{\lambda} =\frac{1}{5}=0.2\\\\ \to Variance =\frac{1}{\lambda^2} =\frac{1}{5^2} =\frac{1}{25}=0.04\\\\$

Let $T = X_1 + X_2 +...+ X_{70}$ , be the total transmission time. According to the Central Limit Theorem, T has a Normal distribution of mean, Variance.

mean $= E(T) = 70 \times 0.2=14$

Variance $=V(T) =70 \times 0.04 = 2.8$

Therefore

$\to P(T$

= P(Z

Therefore, the final answer is "0.1170".

Learn more about the Central Limit Theorem:

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Answer from: sheram2010

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.

Seventy independent messages are sent from an electronic transmission center.Messages are processed

Answer from: Quest

answer: the answer is one of

$f=\dfrac{1}{2},~\dfrac{1}{4},~\dfrac{3}{4},~\dfrac{1}{6},~\dfrac{5}{6},~\dfrac{1}{8},~\dfrac{3}{8},~\dfrac{5}{8},~\dfrac{7}{8}.$

step-by-step explanation: we are given to write a proper fraction where the whole is divided into an even number of equal parts less than ten, and an odd number of those parts exist.

an even number less than 10 may be one of 2, 4, 6 or 8. so, the whole may be one of these four evens.

now, an odd number of these parts must exist, so the numerator must be an odd number, one of 1, 3, 5 or 7. it cannot be 9 because then the fraction will become improper.

thus, our fraction may be one of the following

$f=\dfrac{1}{2},~\dfrac{1}{4},~\dfrac{3}{4},~\dfrac{1}{6},~\dfrac{5}{6},~\dfrac{1}{8},~\dfrac{3}{8},~\dfrac{5}{8},~\dfrac{7}{8}.$

Answer from: Quest

step-by-step explanation:

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