Statistics and Probability Answers

A,B and C are three events. Express the following events in set notations.

 (i) Simultaneous occurrence of A,B and C.

 (ii) Occurrence of at least one of them. 

 (iii) Both A and B occur and C does not occur. 

 (iv) The event B but not A occurs. 

 (v) Not more than one of the events A,B and C occur.

For the given distribution:

P(X=x)=(2/3)(1/3)^x ;x=0,1,2,.....,find moment generating function, mean and variance of X.

(a) For a distribution, the mean is 10, variance is 16, the skewness sk₄ is +1 and kurtosis b₂ is 4. Obtain the first four moments about the origin i.e. zero. Comment upon the nature of the distribution. 

 (b) Find the mean and variance of binomial distribution.

The weights of 1000 children, in average is 61kg with a standard deviation of 18kg. Suppose the weights ate normally distributed, how many children weigh less than 35kg?

For 25 army personnels, line of regression of weight of kidneys (Y) on weight of 

heart (X) is Y=0.399X+6.934 and the line of regression of weight of heart on 

weight of kidney is X−1.212Y +2.461=0. Find the correlation coefficient between 

X and Y and their mean values

Let X_1,X_2,......,X_n be independently and identically distributed b(1, p) random 

variables. Obtain a confidence internal for p using Chebychev’s inequality.

Let X₁,X₂,.....,X_n be random sample of size n from a distribution with probability 

density function 

f(X; ,0)={ (θX) to the power (θ-1) ,0<X<1 , θ>0 Obtain a maximum likeyhood

0, elsewhere.

estimator of θ.

A factory produces steel pipes in three plant with daily production volumes of 500, 

1000 and 2000 units respectively from each of the plants. From the past experience it 

is known that the fraction of defective outputs produced by three plants are 

respectively 0.005, 0.008 and 0.010. If a pipe is selected at random from a day’s total production and founded to be defective, from which plant is that likely to 

have came?

If a random variable u has t -distribution with n degree of freedom, find the 

distribution of u².

For normal distribution with mean zero and variance σ² show that:

E(|x|)= (√2/x)σ