Question

the cases considered in the article). Then X, the number of failures, has a Poisson distribution with μ=1. (Round your answers to three decimal places.) (a) Obtain P(X≤4) by using the Cumulative Poisson Probabilities table in the Appendix of Tables. P(X≤4)= (b) Determine P(X=1) from the pmf formula. P(X=1)=1 Determine P(X=1) from the Cumulative Poisson Probabilities table in the Appendix of Tables. P(X=1)=[ (c) Determine P(1≤X≤3). P(1≤X≤3)= (d) What is the probability that x exceeds its mean value by more than one standard deviation? You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

(a) Using the Cumulative Poisson Probabilities table,
`\( P(X \leq 4) \)` is found to be approximately 0.983.

(b) From the Poisson probability mass function (pmf) formula,
`\( P(X = 1) \)` is calculated as
`\( (e^(-1) \cdot 1^1)/(1!) \),` which equals 0.368. This value is also confirmed by the Cumulative Poisson Probabilities table.

(c) To determine
`\( P(1 \leq X \leq 3) \)`, we subtract
`\( P(X \leq 0) \)` from
`\( P(X \leq 3) \).` Using the table,
`\( P(1 \leq X \leq 3) \)` is approximately 0.647.

(d) The probability that
`\( X \)` exceeds its mean value by more than one standard deviation can be calculated using the properties of the Poisson distribution. First, find the mean
`(\( \mu \))` and standard deviation
`(\( \sigma \))`, and then compute
`\( P(X > \mu + \sigma) \).` This involves using the Cumulative Poisson Probabilities table and relevant formulas.

Step-by-step explanation

(a) The Cumulative Poisson Probabilities table provides the cumulative probabilities for a Poisson distribution. In this case, to find
`\( P(X \leq 4) \),` locate the value in the table corresponding to
`\( X = 4 \)`, resulting in approximately 0.983.

(b) The Poisson probability mass function (pmf) is given by
`\( P(X = k) = (e^(-\mu) \cdot \mu^k)/(k!) \),` where
`\( \mu \)` is the mean. For
`\( P(X = 1) \)`, substitute
`\( k = 1 \)` and
`\( \mu = 1 \)`, yielding 0.368.

(c) To find
`\( P(1 \leq X \leq 3) \)`, subtract the cumulative probability at
`\( X = 0 \)`from the cumulative probability at
`\( X = 3 \).` This accounts for the probability of
`\( X \)` taking values between 1 and 3. The result is approximately 0.647.

(d) The probability that
`\( X \)` exceeds its mean by more than one standard deviation involves determining
`\( P(X > \mu + \sigma) \)`. Calculate the mean
`(\( \mu \))` and standard deviation
`(\( \sigma \))`, then use the Cumulative Poisson Probabilities table to find the desired probability. This requires understanding the relationship between the Poisson distribution parameters and their impact on probabilities.