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A tail of a sum of normals

EasyRandom VariablesStatistics~3m

X∼N(6, 9)X\sim N(6,\,9) and Y∼N(4, 16)Y\sim N(4,\,16) are independent (the second argument is the variance). Roughly what percent of the time does X+YX+Y exceed 2020? Use the empirical rule. (to 1 decimal)

Show hints (2)+
  1. First add the variances (9 + 16 = 25) to get the sum's SD of 5, then standardize: z = (20-10)/5 = 2.
  2. The upper tail beyond +2 SD is about half of the leftover 5%.

Answer

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2.3 (± 0.5)

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Asked at: Timed Mental-Math & Sequences, Game-Based Aptitude

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