1. The conditional probability mass (density) function of X, given Y = y is defined as F for (x, y) = where f()) * 0 and fr() 2. The conditional probability mass (density) function of Y, given X = x is defined as T fxr (x, y) where fr (x) * 0. fx(x)Distribution with Expansions for Test of Hypothesis Test Statistic Degrees of Freedom Notations Difference between means of popu- X - HO *-distribution with X = Sample mean lation and sample (o unknown) v=n- 1 vn- 1 Difference between means of two (X - X2) t-distribution with = Population mean samples ( unknown) t= / 1 v=n, +n- 2 (a) If n, # n n2 (b) Ifn, =n, =n (*1 - X2) f-distribution with s = Sample standard t = v = 20 - 2 deviation n = Sample size V (n- 1) D = Degrees of freedom (c) n, = n, =n and the two samples d t-distribution with t= are not independent i.e., they are v=n-1 related in some way (This implies Vn- 1 that the pairs of observations (x, where d = x, - y y) belong to same sample unit d, = x, - y, and $2 = _ _(d, - a)] Equality of the population variances F-distribution with Note: Take the larger of the esti- F = v =n - 1 mates of variances of the samples Where of = [ ," ) of and a) - [me , )s? v2= n - 1 as of and the corresponding degrees of freedom as Population variance -distribution with v=n- 1 where 8 = (x -x] n\f\f\f\fQ4: An eclge (If a variable cube is increasing at the rate [if 3 cmis. Haw fast is the ucnlume of the cube increasing 1when the edge is 1G cm long? \fQuestion 13: Find the direction cosines of the vector joining the points A (1, 2, 3) and B (1, 2, 1) directed from A to B. \fQuestion 19: If 5 and are two collinear vectors, then which of the following are incorrect: A. 5:35, for some scalar A B. (i = if; C. the respective components of '5 and 5 are proportional D. both the vectors '5 and b have same direction, but different magnitudes \f\f15. Question A binary operation * i5 defined an the Set R of all real numbers by the rule El .-::b = a: * b2 far all a, b E R. Write the identityr element for * an Ft. \f5f 5' ~ 2&7 Find a vector in the direction of vector which has magnitude 8 units. \f\f2. Question On the SET: Z of all integers a binary operation * i5 defined by a * t: = a + b + 2 for all a; b E Z. Write the inverse of 4. \f\f\f\fQuestion ?: Find the unit vector in the direction of the vector";I = t 4'} +2k. \f2|]. Question Let * bE a binary.r operation on set of integers. I, defined by a * b = 23 + t: -3. Find the value of 3 * 4. Proposition 2.3.2. For any constants a, b E R, the expectation on 2 (IX + b is E(aX + b) = aE{X) + b. \fL Question 1: Prove that sum of residuals in any regression model that contains an intercept b is always 0 1) Prove : Dei = _(yi - Di) = 0\f\f\fQuestion 7: Solve the following The PDF of a random variable X is 104:x30 10 0 otherwise What is P(X 1 (x - D) Solution