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Obtaining unknown mean and variance

Obtaining unknown mean and variance

IB Year 6 | Grade 12

A random variable \(Y\) has a normal distribution.

  1. Calculate the mean and the standard deviation, given that \(\mathrm{P}(Y > 5.82) = 0.04\) and \(\mathrm{P}(Y < -3.74) = 0.18\).
  2. The random variable \(R\), also normally distributed, has the same standard deviation as \(Y\), but its mean is unknown. Find the greatest possible value of \(\mathrm{P}(-3.74 < R < 5.82)\).

Solution:

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Answer:(a) Mean \(\mu\approx-0.458\), standard deviation \(\sigma\approx3.59\) (3 s.f.). (b) Maximum probability \(\approx0.817\) (3 s.f.), attained when \(\mathrm{E}(R)=1.04\).

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