Let \(X\) be a random variable uniformly (rectangularly) distributed over the interval \(0 < x < 1\). Derive the probability density functions of the following random variables \((a)\) \(Y = X^2 - 1\), \((b)\) \(Z = \sin\pi X\). Find the mean and standard deviation of \(Y\) and \(Z\).
A manufacturer is asked to supply steel tubing in lengths of 10 feet. Several samples are obtained from him and the mean lengths in feet of four samples each of 16 tubes found to be as follows: $$10 \cdot 16; \quad 10 \cdot 38; \quad 10 \cdot 31; \quad 10 \cdot 07.$$ What type of distribution would you expect mean lengths such as these to have and why? Samples are also obtained from another source, and in this case the mean lengths in feet of five samples of 16 tubes are found to be as follows: $$10 \cdot 15; \quad 10 \cdot 36; \quad 10 \cdot 11; \quad 10 \cdot 11; \quad 10 \cdot 07.$$ Assuming that both manufacturers produce tubing whose length has a standard deviation of \(0 \cdot 48\) feet, is there any evidence that either manufacturer's tubing has a mean length greater than \(10 \cdot 1\) feet? Is there any evidence that tubes supplied by the two manufacturers differ in mean length? [Let $$\Phi(X) = \int_{-\infty}^x \phi(x)dx,$$ where \(\phi(x) = (2\pi)^{-1}\exp(-\frac{1}{2}x^2)\). Then \(\Phi(-2 \cdot 58) = 0 \cdot 005\), \(\Phi(-2 \cdot 33) = 0 \cdot 01\), \(\Phi(-1 \cdot 96) = 0 \cdot 025\), \(\Phi(-1 \cdot 64) = 0 \cdot 05\).]