Transcendental function expectations

Questions involving E(g(X)) where g contains trigonometric, exponential, or other transcendental functions requiring specialized integration techniques or numerical methods.

2 questions

OCR Further Statistics 2021 November Q8
8 The continuous random variable \(Y\) has a uniform distribution on [0,2].
  1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
  2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).
SPS SPS FM Statistics 2022 February Q6
6. The continuous random variable \(Y\) has a uniform distribution on [ 0,2 ].
  1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
  2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).
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