Find the solution to the following questions.
1. If p is the probability of 'successes' at a single trial, obtain the probability of r 'successes' out of n independent trials. 2. Discuss the nature of mode of binomial distribution. 3. Show that if np be a whole number, the mean of the binomial distribution coincides with the greatest term. 4. Obtain the m.g.f. of the binomial distribution. Hence or otherwise obtain the mean, variance and skewness of distribution. 5. State and prove the reproductive property of the Poisson distribution. Show that the mean and variance of the Poisson distribution are equal. Find the mode of the Poisson distribution with mean value 5. 6. Show that for a Poisson distribution, the coefficient of variation is the reciprocal of the standard deviation. 7. If X1, X2. .... Xk are independent random variables following the Poisson law with parameter M1, M2. .... my respectively, show that _=1 X; follows the Poisson law with parameter _ _, mi. 8. Prove the recurrence relation between the moments of Poisson distribution: Hr+1 = 1 (rur-1 + "er), v ax ), where u is the rth moment about the mean A. Hence obtain the skewness and kurtosis of Poisson distribution. 9. Let X has a Poisson distribution with parameter) > 0. If r is a non-negative integer and if ur'= E(X'), prove that #'7+1 = 2 (1, + ) 10. Obtain the Poisson distribution as a limiting case of the negative binomial distribution.QUESTION 1 1 points Save Answer Use the table for the Poisson Distribution to find F(5;6.4) QUESTION 2 1 points Save Answer Use the table for the Poisson distribution to find f(5; 6.4) QUESTION 3 1 points Save Answer Use the table for the Poisson Distribution to find: 13 E f ( k; 9.5) k =8A Type ll error is made when we reject the null hypothesis and the null hypothesis is actually false. when we accept the null hypothesis and the null hypothesis is actually false. when we reject the null hypothesis and the null hypothesis is actually true. when we accept the null hypothesis and the null hypothesis is actually true. 7.7 Refer to exercise 2.3 concerning the distribution of ant lion pits. Using the sample mean as the best estimate of the parametric mean, deter- mine if the distribution of pits follows a Poisson distribution. Interpret your results biologically. (Note: When determining the Poisson distribu- tion carry out x until the cumulative probability equals 1. Use a statistical program to determine the Poisson Distribution.) Number of Quadrats Pits/Quadrat (X) Containing X Pits (Frequency) 5 15 23 DO YOUTAWNHO
