(Solved): 1. Let X be a continuous random variable whose probability density function is f(x)={4x300 ...

1. Let $X$ be a continuous random variable whose probability density function is $f(x)=\{\begin{array}{ll}\frac{x^3}{4}& 0?x?c\\ 0& \text{ otherwise }\end{array}$ What is the value of $c$ that makes $f(x)$ a valid probability density function (PDF). 2. Let $f(x)=\{\begin{array}{ll}0.048x(5?x)& 0?x?5\\ 0& \text{ otherwise }\end{array}$ (a) Verify that $f$ is a probability density function. (b) Find $P(X?4)$. 3. Sketch the region bounded by the given curves, and then find the area of the region. (a) $y=x^2?x,y=3?x^2$ (b) $y^2=x,4y?y^2=x$