Solve the question below kindly
How many chords can be drawn through 21 points on a circle?
4. How many triangles can be formed by joining the vertices of a hexagon?
5. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels
can be formed?
6. If four dice are rolled, find the number of possible outcomes in which atleast one
die shows 2.
7. There are 18 guests at a dinner party. They have to sit 9 guests on either side of
a long table, three particular persons decide to sit on one particular side and two
others on the other side. In how many ways can the guests to be seated?
8. If a polygon has 44 diagonals, find the number of its sides.
9. In how many ways can a cricket team of 11 players be chosen out of a batch of 15
players?
(i) There is no restriction on the selection.
(ii) A particular player is always chosen.
(iii) A particular player is never chosen.
10. A Committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this
can be done when
(i) atleast two ladies are included
(ii) atmost two ladies are include
Neyman-Pearson Lemma. (Lema Neyman-Pearson.) [2 marks] (b) Assume that scores of the students, Y for a mathematics course are normally distributed with unknown mean and variance, 30. We wish to test null hypothesis Hail = 75 against alternative hypothesis H,: 1 = 77. Let C = [Y : Y > 76.8) be critical region of the test, where Y is the observed mean sample. Compute the power of test if n = 50. [6 marks ] c) Let X,, X,..... X denote a random sample of size n from a Poisson distribution with mean A. Obtain the likelihood ratio critical region for testing Ho: 1 = 2 vs Hid = 5. [5 marks] "t X X,.....X, be a random sample from a normal distribution with unknown mean, and variance, o' = 0.16. We wish to test Ho : 0 = 5.0 versus H : 0 = 5.5.3.1. Consider the standard Hamiltonian for a system of NV identical particles H = CPE + U( 1 1, . , IN). 1=1 a. Show that the microcanonical partition function can be expressed in the form S ( N , V, E ) = MN / dE' Japs ( Z E' 2m * dor 6 (U ( F 1 . . . . . IN) - E + E') , which provides a way to separate the kinetic and potential contributions to the partition function. b. Based on the result of part a, show that the partition function can, there- fore, be expressed as Eo 2 Tm 3/27 N n(N, V, E) = NIT (3N) h2 JD(V) der [E- U(ri, .... rx)] 13N/2-10 (E- U(F1, ..., IN)) , where O(x) is the Heaviside step function.Let X1, ..., Xn be i.i.d. samples from Poisson(1). We want to test the null hypothesis Ho : 1=2 against the alternative hypothesis H1 : 1 =5. (i) Compute the ratio L(2)/L(5) of joint likelihood functions. Show that L(2) s k L(5) xi log(2/5) + 3n s log k. (ii) Using the Neyman-Pearson lemma, conclude that a best critical region C for testing Ho : 1 = 2 against H1 : 1 =5 is given by C =(x1, . . ., Xn) Exizcy. i=1 (iii) Let n = 30. Use CLT and (ii) and obtain a best critical region C for testing Ho : 1 = 2 against H1 : 1 =5 at significance level a = 0.05.1. Distribution and Moments of Jointly distributed Random Variables Topics: Probability distribution discrete and continuous RV Moments: Mean, Variance Joint Probability distribution discrete and continuous RV's Marginal distribution discrete and continuous RV Conditional Probability distribution discrete and continuous RV's Moments: conditional, unconditional, joint monents Value of Y Probability Distribution 14 22 30 40 65 of X Value of X 0.02 0.05 0.10 0.03 0.01 0.17 0.15 0.05 0.02 0.01 0.02 0.03 0.15 0.10 0.09 Probability distri- bution of Y a) What is the marginal distribution (probability distribution) of X and the marginal distribution (probability distribution) of Y? Draw a graph b) Calculate the Expected Value of X and the Expected value of Y? c) Calculate Var(X) and Var(Y) d) Calculate E(Y X=8) e) Calculate Var(Y X=8) f) Calculate Cov(X, Y) g) Are X and Y independently distributed
