For normal distribution with mean zero and variance σ2 show that:
We are asked to show that for a normal distribution with mean 0 and variance $\sigma^2$: $$ E(|X|) = \sqrt{\frac{2}{\pi}} \, \sigma $$ Let's solve this step by step manually. --- ### Step 1: Write the expectation formula For a continuous random variable $X \sim N(0, \sigma^2)$: $$ f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-x^2/(2\sigma^2)} $$ The expected value of $|X|$ _____ ________ ________ _____ _____ ________ _______ _______ ___.
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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 the given distribution:
find moment generating function, mean and variance of X.
Let X1, X2,,,,,,, Xn , , , be independently and identically distributed b ,(1,p) random variables. Obtain a confidence internal for p using Chebychev’s inequality.
Let X be a binomial variate with n =100, p = .1.0 Find the approximate value of P (10 ≤ X ≤12) using:
(i) normal distribution
(ii)poisson distribution
[You may like to use the following values.
P(Z ≤ 67.0 ) = .0 7486, P(Z ≤ 0.33 ) = .0 6293, P(Z ≤ 0 ) = ]5.0
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.
(a) Poisson distribution is a limiting case of binomial distribution for and
(b) For two independent events A and B, if P(A) = 2.0 and P(B) = ,4.0 then (A ∩ B) = 0.6.
(c) If H0 : P ≤ 6.0 and X ~ B(n, p)n -known and p unknown and H1 :µ = µ0 where X ~ N (µ,σ2 )σ2 unknown, then H0 and H1 are simple null hypothesis.
(d) Frequency density of a class for any distribution is the ration of total frequency to class width.
(e) If X and Y are independent r.v.s with MX (t) and My (t) as their m.gf’s respectively, then Mx+y (t) Mx (t)My (t).