How to Solve Discrete Random Variables — Univariate RV — Poisson distribution Questions on Exam P
Sample Practice Problem
ID: #D6Q4F
Let \(S\) be the combined count of independent events; it is Poisson with mean 60. Using a normal approximation with continuity correction, approximate the probability the total is between 52 and 54 inclusive.
📖 Worked Solution & Strategy
The Poisson total \(S\) has mean \(\mu = 60\) and variance \(\sigma^2 = 60\). We approximate \(S\) by a normal distribution \(S \approx \operatorname{Normal}(60, 60)\). Continuity correction converts integer bounds \(k\) into continuous intervals \([k-0.5, k+0.5]\). For inclusive interval \(52 \le S \le 54\), continuity correction uses \([51.5, 54.5]\). Standardizing gives \(z_L = \frac{51.5 - 60}{\sqrt{60}} = -1.0973\) and \(z_U = \frac{54.5 - 60}{\sqrt{60}} = -0.7100\). The probability is \(\Phi(z_U) - \Phi(z_L) = \Phi(-0.7100) - \Phi(-1.0973) = 0.1026\). The evaluated result is \(0.1026\), which matches the correct option.