(Solved): part b Suppose a certain type of small data processing firm is so specialized that some have difficu ...

Suppose a certain type of small data processing firm is so specialized that some have difficulty making a profit in thair first year of operation. The probablity density function that characterizes the proportion \( Y \) that make a profit is given. \[ f(y)=\left\{\begin{array}{rr} k y^{7}(1-y)^{3}, & 0 \leq y \leq 1 \\ 0, & \text { etsewhere } \end{array}\right. \] Complete parts (a) through (c). (a) What is the value of \( k \) that renders the above a valid density function? \( k=\quad \) (Type an integer or a fraction.) (b) Find the probability that at most \( 52 \% \) of the firms make a profit in the first year. (Round to four decimal places as needed.)