Question

{"given":"Random variable $X$ follows Poisson distribution with mean $\\lambda = 2$. We need to find $P(X > 1.5)$. For Poisson distribution, $P(X = k) = \\frac{e^{-\\lambda} \\lambda^k}{k!}$ where $k$ takes non-negative integer values.","key_observation":"Since $X$ is a discrete random variable that can only take integer values (0, 1, 2, 3, ...), the event $X > 1.5$ is equivalent to $X \\geq 2$. This is because there are no possible values between 1 and 2 for a discrete distribution. We can use the complement rule: $P(X \\geq 2) = 1 - P(X < 2) = 1 - P(X = 0) - P(X = 1)$.","option_analysis":[{"label":"(A)","text":"Option contains an image","verdict":"incorrect","explanation":"Option contains an image. Based on the Poisson calculation, the answer should be $1 - 3e^{-2}$, which is approximately 0.594. Without seeing the image content, this is likely not the correct mathematical expression."},{"label":"(B)","text":"0","verdict":"incorrect","explanation":"This is incorrect. $P(X > 1.5) = P(X \\geq 2) \\neq 0$ since there is a positive probability that $X$ takes values 2, 3, 4, etc. The probability is actually $1 - 3e^{-2} \\approx 0.594$."},{"label":"(C)","text":"Option contains an image","verdict":"correct","explanation":"Option contains an image. Based on our calculation, $P(X > 1.5) = P(X \\geq 2) = 1 - 3e^{-2}$. Since this is the stored answer and matches our mathematical derivation, this image likely contains the expression $1 - 3e^{-2}$."},{"label":"(D)","text":"Option contains an image","verdict":"incorrect","explanation":"Option contains an image. Since option C is correct with the expression $1 - 3e^{-2}$, this option likely contains a different mathematical expression that does not equal the correct probability value."}],"answer":"(C)","formula_steps":[]}