n 5.6.4. Show that ô² = ΣY? is sufficient for o² if i=1 Y1, Y2,..., Yn is a random sample from a normal pdf with μ = 0.
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- EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.1. 2. 3. Let Y₁, Y2, Y3, Y4, Y5 be a random sample of size 5 from a standard normal population. Find the moment generating function of the statistic: X = 2Y₁²+ Y₂² + 3Y3² +Y4² +4Y5² Let Y₁, Y₂, Y3, Y4, Y5 and X₁, X₂,..., X, be independent and normally distributed random samples from populations with means ₁ = 2 and ₂ = 8 and variances ₁² = 5 and ₂² = k, respectively. Suppose that P(Ỹ – Ỹ > 10) = 0.02275, find the value of ₂² = k. Suppose that Y₁, Y₂2, ..., Ym and X₁, X2, Xm are independent normally distributed random samples from populations with means μ₁ and μ₂ and variances ₁² and ₂², respectively. Is X - Y a consistent estimator of ₂ - ₁? Justify your answer.
- obtain the MVB estimator for u in normal population №.(482), where 2 is know2.5.1. Show that the random variables X₁ and X₂ with joint pdf ƒ(7₁, 72) = { 1271²2(1-22) 0<+1 <1,0<2₂ <1 0 elsewhere are independent.2.2 Let the variable X have the binomial distribution, that is, p(x) = (") p*(q)"-*, x= 0,1,2, -- ,n If the mean of X is given to be µ = np. Find E[X(X – 1)]. Use results in (i) to find the variance of X. i. ii.
- Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=12.4. Let Y,, Y2, .., Yn denote a random sample from a uniform distribution with a pdf given by f(y|e) = }0 09.1.4 X1, X2 and X, are iid continuous uniform random variables. Random variable Y = X, + X2 + X, has expected value E[Y] = 0 and variance of = 4. What is the PDF fx,(x) of X,?
