Help me to solve the following statistical questions. Thanks

A data set of 20 observations has mean 45 and standard deviation 25.4. The data set is reviewed and one observation which was incorrectly recorded as 130 is now corrected to 30. Determine the mean and standard deviation of the corrected data. [4] A random variable, X', has the probability generating function G,(n) where Gx() -0.4096 +0.4096 + 0.1528/ - 0.02567 + 0.00161 (i) Determine the probability P(X" -3) using G,(n). [1] You are now given that X follows a binomial distribution. (ii) Determine the parameter values of the distribution of X. [3] [Total 4] A sports scientist is building a statistical model to describe the number of attempts a high jump athlete will have to make until she succeeds in clearing a certain height for the first time during an indoor sports event. For this model the scientist considers a geometric distribution with probability of success p. The cumulative distribution function of the geometric distribution is given as Fx() =1-(1-p)', x=1, 2,3, ... (a) State the assumptions that the scientist needs to make for considering this distribution. (b) Comment on the validity of the assumptions in part (i)(a). [3] The athlete has tried # jumps without success. (ii) (a) Determine the probability that the athlete will require more than x additional jumps to succeed in clearing the height (b) Comment on what the answer in part (ii)(a) means for the athlete. [31 [Total 6]One card is picked from an ordinary pack of 52 playing cards. Consider the following events A = [pick a 7), B = [pick an ace] and C = [pick a club). Show that Axiom 3, namely P(AUB) = P(A) + P(B): (i) works for events A and B (ii) doesn't work for events A and C and explain why