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Answer :
The answer to the first question is D) 0.036. In a Poisson distribution, the probability mass function gives the probability of a certain number of events occurring in a fixed interval of time or space. It is defined by the average number of events (denoted by λ). In this case, we are given a specific relationship between the probabilities P(X = 3) and P(X = 4).
To solve the problem, we are given that the probability of X being equal to 3 is twice the probability of X being equal to 4. In a Poisson distribution, the probability mass function is given by P(X = k) = (e^(-λ) * λ^k) / k!, where λ is the average number of events.
Let's denote the probability of X being equal to 3 as P(X = 3) = p. Therefore, P(X = 4) = p/2.
We can set up the equation as follows: p = 2 * (p/2) * (e^(-2)) / 4!
Simplifying this equation, we get: p = (p * e^(-2)) / 12
Multiplying both sides by 12, we obtain: 12p = p * e^(-2)
Dividing both sides by p, we have: 12 = e^(-2)
To find P(X = 5), we can substitute the value of λ = 2 into the Poisson probability mass function:
P(X = 5) = (e^(-2) * 2^5) / 5!
Calculating this expression, we get P(X = 5) ≈ 0.036, which corresponds to option D.
We can set up an equation by substituting the given probabilities into the Poisson probability mass function. Simplifying the equation, we find that the probabilities are related by the exponential term e^(-2).
To find P(X = 5), we substitute the average number of events λ = 2 into the probability mass function. After calculating the expression, we obtain the value of approximately 0.036.
Therefore, the answer to the first question is option D) 0.036.
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