(Solved): math  5. Calculate the first four central moments for the following frequency distribution. 6. ...

5. Calculate the first four central moments for the following frequency distribution. 6. A bag contains 16 identical balls of which 10 are white and the rest are black. Two balls are drawn at random from the bag. What is the probability that both balls are black? 7. A bag contains 5 white and 6 red halls. Two halls are drawn successively at random from the bag. What is the probability that both the balls are white when the drawings are made (i) with replacement (ii) without replacement? 8. There are two identical boxes containing respectively 4 white and 3 red balls, 3 white and 7 red balls. A box is chosen at random and a ball is drawn at random from it. If the ball is white, what is the probability that it is from the second box? 9. The probability density function of a random variable \( x \) is given below: \[ \begin{aligned} f(x) & =k x, & & 02) \). 10. Prove that the variance of a binomial distribution is \( n p q \). 11. An unbiased coin is tossed 10 times. Find the probability of getting (i) 3 heads (ii) at least 1 head (iii) at most 1 head. 12. A factory produces blades in packets of 10. The probability of a blade to be defective is \( 0.2 \% \). Find the number of packets having two defective blades in a consignment of 10,000 packets. 13. The following table gives the aptitude test scores and productivity indices of 10 workers selected at random Estimate (i) The productivity index of a worker whose test score is 92. (ii) The test score of a worker whose productivity index is 75 .