Question

If \(6\) is the mean of a Poisson distribution, then
\[ P(X\geq3)= \]

• A. \(1-\dfrac{25}{e^6}\)
• B. \(e^{-6}-25\)
• C. \(24-25e^6\)
• D. \(e^{-3}\)

Hint:

For Poisson distribution, use the complement rule for probabilities like \(P(X\geq r)\) whenever calculating smaller terms is easier.

Answer 1

The Correct Option is A

Step 1: Identify the parameter of the Poisson distribution.
For a Poisson distribution, mean is equal to the parameter \(\lambda\).
Given, mean \(=6\).
Therefore,
\[ \lambda=6 \]

Step 2: Use the complement rule.
We need to find
\[ P(X\geq3) \]
Using complement rule,
\[ P(X\geq3)=1-P(X\lt 3) \]
\[ =1-[P(X=0)+P(X=1)+P(X=2)] \]

Step 3: Use the Poisson probability formula.
For a Poisson random variable,
\[ P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!} \]
Since \(\lambda=6\),
\[ P(X=0)=\frac{e^{-6}6^0}{0!}=e^{-6} \]
\[ P(X=1)=\frac{e^{-6}6^1}{1!}=6e^{-6} \]
\[ P(X=2)=\frac{e^{-6}6^2}{2!} \]
\[ =\frac{36}{2}e^{-6} \]
\[ =18e^{-6} \]

Step 4: Add the probabilities.
\[ P(X\lt 3) = e^{-6}+6e^{-6}+18e^{-6} \]
\[ =25e^{-6} \]
Thus,
\[ P(X\geq3)=1-25e^{-6} \]
Since
\[ 25e^{-6}=\frac{25}{e^6}, \] we get
\[ P(X\geq3)=1-\frac{25}{e^6} \]

Step 5: Final conclusion.
Hence,
\[ \boxed{1-\frac{25}{e^6}} \]