Final answer:
Choosing the appropriate Poisson probability mass function for a distribution with μ=3 involves using the formula P(X=x) = (μ^x * e^-μ)/x!, where x is the number of occurrences and μ is the mean. You can calculate specific probabilities for exact values, ranges, and for X greater than or equal to a certain number, ensuring to round to four decimal places.
Step-by-step explanation:
To choose the appropriate Poisson probability mass function for a distribution with μ=3, you need to understand that the Poisson distribution is used for modeling the number of times an event happens within a specific interval of time or space when the events are independent. Here, μ represents the average number of occurrences (mean) in the interval. For instance, in the case of a lightbulb factory where the average number of defective bulbs (μ) is 3, the distribution would be denoted as X~P(3), meaning that the random variable X follows a Poisson distribution with a mean of 3.
The Poisson probability mass function is given by:
P(X=x) = (μ^x * e^-μ)/x!
where e is the base of the natural logarithm, and x! is the factorial of x. Here's how you can calculate specific probabilities:
- To find P(X = 3), you would plug 3 into the function: P(X=3) = (3^3 * e^-3)/3!.
- For a range, like P(1 < X < 4), you would calculate for each value of X and then sum them up.
- To find P(X ≥ 2), you would sum the probabilities of all occurrences from 2 onwards.
Remember to round your answers to four decimal places for precision. If μ equals np, where n is the number of trials and p is the probability of success, you have an approximation for the binomial distribution using the Poisson distribution, which is valid when n is large and p is small.