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1. Let \( X \) have a Poisson distribution with a mean of \(\mu = 5\). The value of \( P(X = 6) \) is:
A. 0.123
B. 0.132
C. 0.146
D. 0.152

2. Let \( X \) have a Poisson distribution with a mean of \(\mu = 5\). The value of \( P(X > 5) \) is:
A. 0.384
B. 0.143
C. 0.843
D. 0.652

3. Let \( X \) have a Poisson distribution with a variance of \( V(X) = 3 \). The value of \( P(X = 2) \) is:
A. 0.352
B. 0.224
C. 0.412
D. 0.513

4. A screen, on average, has one flaw in a 100-foot roll. What is the probability that a 50-foot roll has no flaw?
A. 0.5041
B. 0.321
C. 0.456
D. 0.6065

5. The mean of a Poisson random variable \( X \) is \(\mu = 9\). What is \( P(\mu - 20 < X < \mu + 20) \)?
A. 0.865
B. 0.957
C. 0.634
D. 0.735

6. The number of knots in a particular type of wood has a Poisson distribution with an average of 1.5 knots in 10 cubic feet of wood. What is the probability that a 10-cubic-foot block of the wood has at most one knot?
A. 0.5578
B. 0.6687
C. 0.7723
D. 0.8812

7. Let \( X \) denote a random variable that has a Poisson distribution with a mean \(\mu = 2\). What is \( P(X > 4) \)?
A. 0.214
B. 0.143
C. 0.345
D. 0.457

Answer :

1. The probability of P(X = 6) is approximately 0.146.

2. The value of P(X > 5) is approximately 0.384.

3. The value of P(X = 2) is approximately 0.224.

4. The probability that a 50-foot roll has no flaw is approximately 0.6065.

5. The probability P(u – 20 < X < u + 20) is approximately 0.957.

6. The probability that a 10-cubic-foot block of the wood has at most one knot is approximately 0.5578.

7. The value of P(X > 4) is approximately 0.457.

1. The Poisson probability mass function is given by:

[tex]P(X = k) = (e^{-u} * u^k) / k![/tex]

For u = 5 and k = 6:

[tex]P(X = 6) = (e^{-5} * 5^6) / 6! = 0.146[/tex]

So, the answer is (C) 0.146.

2. The Poisson distribution is memoryless, which means P(X > 5) = 1 - P(X ≤ 5):

P(X > 5) = 1 - P(X ≤ 5)

[tex]= 1 - \sum(k=0 to 5) [(e^{-5} * 5^k) / k!][/tex]

≈ 0.384

So, the answer is (A) 0.384.

3. The variance of a Poisson distribution is V(X) = u. For u = 3 and k = 2:

[tex]P(X = 2) = (e^{-3} * 3^2) / 2! = 0.224[/tex]

So, the answer is (B) 0.224.

4. The Poisson parameter for a 50-foot roll is u = 50 / 100 = 0.5. For k = 0:

[tex]P(X = 0) = (e^{-0.5} * 0.5^0) / 0! = 0.6065[/tex]

So, the answer is (D) 0.6065.

5. The mean of a Poisson distribution is u = 9. To find P(u - 20 < X < u + 20), we can use the normal approximation to the Poisson distribution:

P(u - 20 < X < u + 20) ≈ P(-2 < Z < 2), where Z is a standard normal random variable.

Using the standard normal distribution table, P(-2 < Z < 2) ≈ 0.9545.

So, the answer is approximately (B) 0.957.

6. The parameter for a 10-cubic-foot block is u = 1.5. For k = 0 and k = 1:

P(X ≤ 1) = P(X = 0) + P(X = 1)

[tex]= (e^{-1.5} * 1.5^0) / 0! + (e^{-1.5} * 1.5^1) / 1![/tex]

≈ 0.5578

So, the answer is (A) 0.5578.

7. For u = 2:

P(X > 4) = 1 - P(X ≤ 4)

[tex]= 1 - \sum(k=0 to 4) [(e^{-2} * 2^k) / k!][/tex]

≈ 0.457

So, the answer is (D) 0.457.

To know more about probability , refer here:

https://brainly.com/question/32117953

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