(Solved): Example 5-16 The amount of bread (in hundreds of pounds) \( x \) that a certain bakery is able to ...

Example 5-16 The amount of bread (in hundreds of pounds) \( x \) that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the p.d.f. \( f(x) \), given by : \[ f(x)=\left\{\begin{array}{cc} k \cdot x, & \text { for } 0 \leq x<5 \\ k \cdot(10-x), & \text { for } 5 \leq x<10 \\ 0, & \text { otherwise } \end{array}\right. \] (a) Find the value of \( k \) such that \( f(x) \) is a probability density function. (b) What is the probability that the number of pounds of bread that will be sold tomorrow is: (i) more than 500 pounds, (ii) less than 500 pounds, and (iii) between 250 and 750 pounds? (c) Denoting by \( A, B \) and \( C \) the events that the pounds of bread sold are as in \( b \) (i), \( b \) (ii) and \( b \) (iii) respectively, find \( P(A \mid B), P(A \mid C) \). Are (i) \( A \) and \( B \) independent events ? (ii) Are \( A \) and \( C \) independent events ? Example 5.17. A random variable \( X \) has the probability law : \[ d F(x)=\frac{x}{b^{2}} \cdot e^{-x^{2} / 2 b^{2}} d x, 0 \leq x<\infty \] Find the distance between the quartiles and show that the ratio of this distance to the standard deviation of \( X \) is independent of the parameter ' \( b \) '. If the density function of a random variable \( X \) is given by \[ f(x)=k x(1-x), \quad 0 \leq x \leq 1 \] find (i) \( k \), (ii) \( A M \), (iii) HM, (iv) Median, (v) Mode, (vi) \( S D \), (vii) MD about the mean.