(Solved): The actual tracking weight of a record player stylus cartridge that is set to track at \( 3 \mathr ...

The actual tracking weight of a record player stylus cartridge that is set to track at \( 3 \mathrm{~g} \) on a particular player can be regarded as a continuous random variable \( \mathrm{X} \) with pdf \[ f(x)=\left\{\begin{array}{ll} k\left[1-(x-3)^{2}\right] & 2 \leq x \leq 4 \\ 0 & \text { otherwise } \end{array},\right. \] where \( k \) is a positive constant. (a) Sketch the pdf. State the value of \( x \) at which this function is symmetric. (b) Find the value of \( k \) [ 3 points \( ] \). (c) What is the probability that the actual tracking weight is greater than the prescribed weight? (d) What is the probability that the actual weight is within \( 0.25 \mathrm{~g} \) of the prescribed weight? [ 2 points] (e) What is the probability that the actual weight differs from the prescribed weight by more than \( 0.5 \mathrm{~g} \) ? [ 2 points] (f) What is the expected actual tracking weight of the stylus?