Question 1

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.9 , t = 3 , P ( \text { Less than } 3 )$

Accepted Answer

To calculate this probability we can use the cumulative distribution function (CDF) of a Poisson random variable with parameter $\lambda$, which is given by:
$$F(k;\lambda)=\sum_{i=0}^ke^{-\lambda}\frac{\lambda^i}{i!}$$
This represents the probability of observing at most k events in a unit of time or space, when events occur at a rate of $\lambda$ per unit.

In this case, we have $\lambda=0.9$ and $t=3$, which means that the expected number of events in 3 units of time or space is $\mu=\lambda t=2.7$.

We want to calculate $P(X<3)$, which is equivalent to $P(X\leq 2)$, since X is a discrete random variable. Using the Poisson CDF, we have:
$$P(X\leq 2)=F(2;\mu)=\sum_{i=0}^2e^{-\mu}\frac{\mu^i}{i!}=e^{-\mu}\left(1+\mu+\frac{\mu^2}{2}\right)\approx 0.4936$$
Therefore, the best choice is C, with a probability of approximately 0.4936.

Question 2

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\sigma _ { x }$

Accepted Answer

The answer of The number of typographical errors in a...

Question 3

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.6 , t = 7 , P ( \text { No more than } 3 )$

Accepted Answer

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = e^(-0.6) * 0.6^0/0! + e^(-0.6) * 0.6^1/1! + e^(-0.6) * 0.6^2/2! + e^(-0.6) * 0.6^3/3! = 0.3954

Question 4

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 3 pages. Find P(Less than 3).

Accepted Answer

The probability of less than 3 errors on 3 pages is the sum of the probabilities of 0, 1, and 2 errors. Using the Poisson distribution with a mean of 2 errors per page, we have:P(X = 0) = e^(-2) * 2^0 / 0! = 0.1353P(X = 1) = e^(-2) * 2^1 / 1! = 0.2707P(X = 2) = e^(-2) * 2^2 / 2! = 0.2707Therefore, P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767.

Question 5

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 4 per milliliter. What is the probability that there will be Exactly 8 bacteria in a 4-milliliter sample?

Accepted Answer

This is a binomial distribution problem, where we have a fixed number of trials (4 milliliters) and each trial can result in success (presence of bacteria) or failure (absence of bacteria). The probability of success in each trial is given as 4 bacteria per milliliter, which means the probability of failure is 1 - (4/1000) = 0.996.

To find the probability of getting exactly 8 bacteria in a 4-milliliter sample, we can use the following formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

where:
n = number of trials (4 milliliters)
p = probability of success in each trial (4/1000 = 0.004)
k = number of successes we want (8)

Using this formula, we get:

P(X = 8) = (4 choose 8) * 0.004^8 * 0.996^(4-8)
           = (4 choose 8) * 0.004^8 * 0.996^-4
           = (4! / (8! * (4-8)!)) * 0.004^8 * 0.996^-4
           = 0.0120

Therefore, the probability of obtaining exactly 8 bacteria in a 4-milliliter sample is 0.0120, which corresponds to choice B.

Question 6

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 2 per milliliter. Find the standard deviation of the number of Bacteria in a 4-milliliter sample.

Accepted Answer

We know that the standard deviation formula is:

standard deviation = square root of [(Σ(x - μ)^2)/N]

Where Σ is the sum, x is the individual value, μ is the mean, and N is the sample size.

In this case, we want to find the standard deviation of the number of bacteria in a 4-milliliter sample, which is a larger sample than the original one. We can use the formula:

standard deviation = square root of [(Σ(x - μ)^2)/N]

where x = number of bacteria in each milliliter, μ = 2 (since that's the concentration of bacteria in the original sample), and N = 4 (since that's the size of our new sample).

Let's first calculate the sum of (x - μ)^2:

(0 - 2)^2 + (2 - 2)^2 + (4 - 2)^2 + (6 - 2)^2 = 16

Now we can plug this into the formula:

standard deviation = square root of [(Σ(x - μ)^2)/N] = square root of (16/4) = square root of 4 = 2

Therefore, the answer is (C) 2.

Question 7

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(5).

Accepted Answer

Since the number of errors on a page follows a Poisson distribution with mean $\lambda=2$, the number of errors on 4 pages also follows a Poisson distribution with mean $\lambda=8$. Thus, $X \sim 	ext{Poisson}(\lambda=8)$.

To find $P(5)$, we can use the Poisson probability mass function:

$$P(X=5) = \frac{e^{-8}8^5}{5!} \approx 0.0916$$

Therefore, the best choice is B.

Question 8

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\mu _ { x ^ { * } }$

Accepted Answer

The mean of a Poisson distribution is given by $\lambda$, which in this case is 4 errors per page. For 2 pages, the mean number of errors ($\mu _ { x ^ { * } }$) would be $2 \times 4 = 8$, since the number of errors on each page is independent and the mean of the sum of independent Poisson distributions is the sum of their means.

Question 9

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(Greater than 1).

Accepted Answer

Since the number of errors on a page follows a Poisson distribution with mean 2, the number of errors on 4 pages follows a Poisson distribution with mean 8. 
P(Greater than 1) = 1 - P(0 or 1 error) = 1 - (P(X=0) + P(X=1))
Using the Poisson distribution formula with lambda=8:
P(X=0) = e^(-8) * 8^0 / 0! = 0.000335
P(X=1) = e^(-8) * 8^1 / 1! = 0.002688
Therefore, P(Greater than 1) = 1 - (0.000335 + 0.002688) = 0.9970.

Question 10

Last year, a manufacturer produced 200,000 DVD players. Of these, approximately 2% were defective. Assume that a simple random sample of n = 130 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the mean number of DVD players that were Defective.

Accepted Answer

The mean number of defective DVD players ($\lambda$) in a sample of 130 can be found using the Poisson approximation to the binomial distribution. Given that 2% of 200,000 DVD players were defective, the expected number of defective players in the entire lot is 0.02 * 200,000 = 4,000. For a sample of 130, the mean ($\lambda$) is (130/200,000) * 4,000 = 2.6.

Question 11

Last year, a manufacturer produced 1,950,000 DVD players. Of these, approximately 5% were defective. Assume that a simple random sample of n = 230 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the probability that exactly nineteen of the 230 DVD players were defective.

Accepted Answer

The Poisson approximation to the binomial distribution is given by:P(X = k) = (e^-λ * λ^k) / k!where λ = npIn this case, n = 230 and p = 0.05, so λ = 11.5P(X = 19) = (e^-11.5 * 11.5^19) / 19! = 0.0119

Question 12

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 6 per milliliter. Find the mean number of bacteria in a 8-milliliter sample.

Accepted Answer

To find the mean number of bacteria in an 8-milliliter sample, we can use the formula: mean = (concentration) x (volume)

In this case, the concentration is 6 per milliliter and the volume is 8 milliliters.

Mean = (6 per milliliter) x (8 milliliters) = 48 bacteria

Therefore, the mean number of bacteria in an 8-milliliter sample is 48, making choice (B) the correct answer.

Question 13

Last year, a manufacturer produced 350,000 DVD players. Of these, approximately 1% were defective. Assume that a simple random sample of n = 140 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the standard deviation of the number of DVD Players that were defective.

Accepted Answer

First, we need to find the mean of the number of DVD players that are defective in a sample of 140.

Mean (µ) = np = 140*0.01 = 1.4

Next, we need to calculate the variance using the Poisson approximation:

Variance (σ^2) = npq = np(1-p) = 140*0.01*0.99 = 1.386

Finally, we take the square root of the variance to get the standard deviation:

Standard deviation (σ) = √(1.386) ≈ 1.18

Therefore, the closest option in the answer choices is D) 1.2.

Question 14

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.8 , t = 4 , P ( \text { At least } 2 )$

Accepted Answer

The Poisson probability of at least 2 events is 1 minus the probability of 0 or 1 event. Calculate $P(X \geq 2) = 1 - (P(X = 0) + P(X = 1))$ using $\lambda t = 3.2$.

Question 15

Last year, a manufacturer produced 1,950,000 DVD players. Of these, approximately 4% were defective. Assume that a simple random sample of n = 180 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the probability that fewer than four of the 180 DVD players were defective.

Accepted Answer

The answer of Last year, a manufacturer produced 1,950,000 DVD...

Question 16

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.8 , t = 5 , P ( \text { More than } 3 )$

Accepted Answer

The probability of a Poisson random variable being more than a certain value is found by subtracting the cumulative probability of it being less than or equal to that value from 1. Here, we calculate $P(X > 3)$ for a Poisson distribution with $\lambda t = 0.8 \times 5 = 4$. Using a Poisson probability table or calculator, we find $P(X \leq 3)$ and subtract from 1 to get $P(X > 3) = 1 - P(X \leq 3)$. The correct answer, after performing the calculation, is closest to 0.5665.

Quiz 6: Discrete Probability Distributions

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.9 , t = 3 , P ( \text { Less than } 3 )$

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\sigma _ { x }$

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.6 , t = 7 , P ( \text { No more than } 3 )$

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 3 pages. Find P(Less than 3) .

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 4 per milliliter. What is the probability that there will be Exactly 8 bacteria in a 4-milliliter sample?

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 2 per milliliter. Find the standard deviation of the number of Bacteria in a 4-milliliter sample.

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(5) .

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\mu _ { x ^ { * } }$

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(Greater than 1) .

Last year, a manufacturer produced 200,000 DVD players. Of these, approximately 2% were defective. Assume that a simple random sample of n = 130 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the mean number of DVD players that were Defective.

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 6 per milliliter. Find the mean number of bacteria in a 8-milliliter sample.

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.8 , t = 4 , P ( \text { At least } 2 )$

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.8 , t = 5 , P ( \text { More than } 3 )$

Quiz 6: Discrete Probability Distributions

Determine the indicated probability for a Poisson random variable with the given values of λ and t. \lambda \text { and } t \text {. }λ and t. λ=0.9,t=3,P( Less than 3) \lambda = 0.9 , t = 3 , P ( \text { Less than } 3 ) λ=0.9,t=3,P( Less than 3)

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find σx\sigma _ { x }σx​

Determine the indicated probability for a Poisson random variable with the given values of λ and t\lambda \text { and } tλ and t λ=0.6,t=7,P( No more than 3) \lambda = 0.6 , t = 7 , P ( \text { No more than } 3 ) λ=0.6,t=7,P( No more than 3)

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 3 pages. Find P(Less than 3) .

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 4 per milliliter. What is the probability that there will be Exactly 8 bacteria in a 4-milliliter sample?

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 2 per milliliter. Find the standard deviation of the number of Bacteria in a 4-milliliter sample.

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(5) .

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find μx∗\mu _ { x ^ { * } }μx∗​

The number of typographical errors in a document follows a Poisson distribution with a mean of 2 errors per page. Let X represent the number of errors on 4 pages. Find P(Greater than 1) .

Last year, a manufacturer produced 200,000 DVD players. Of these, approximately 2% were defective. Assume that a simple random sample of n = 130 players is drawn. Use the Poisson Approximation to the binomial distribution to compute the mean number of DVD players that were Defective.

An environmental scientist obtains a sample of water from an irrigation canal that contains a certain type of bacteria at a concentration of 6 per milliliter. Find the mean number of bacteria in a 8-milliliter sample.

Determine the indicated probability for a Poisson random variable with the given values of λ and t. \lambda \text { and } t \text {. }λ and t. λ=0.8,t=4,P( At least 2) \lambda = 0.8 , t = 4 , P ( \text { At least } 2 ) λ=0.8,t=4,P( At least 2)

Determine the indicated probability for a Poisson random variable with the given values of λ and t\lambda \text { and } tλ and t λ=0.8,t=5,P( More than 3) \lambda = 0.8 , t = 5 , P ( \text { More than } 3 ) λ=0.8,t=5,P( More than 3)

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.9 , t = 3 , P ( \text { Less than } 3 )$

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\sigma _ { x }$

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.6 , t = 7 , P ( \text { No more than } 3 )$

The number of typographical errors in a document follows a Poisson distribution with a mean of 4 errors per page. Let X represent the number of errors on 2 pages. Find $\mu _ { x ^ { * } }$

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.8 , t = 4 , P ( \text { At least } 2 )$

Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t$ $\lambda = 0.8 , t = 5 , P ( \text { More than } 3 )$