Q1.
() (a) For the sequence of r.v.'s {Xn} with P{Xk = k^λ} = P{Xk = -k^λ} = 1/2, λ > 0. Hence show that if the WLLN holds, central limit theorem holds. r.v.'s {Xn} के अनुक्रम के साथ P{Xk = k^λ}=P{Xk = -k^λ}=1/2, λ > 0.
()
1. Answer the following questions: 6 × 5 = 30\n(a) For the sequence of r.v.'s Xn with PXk = k^λ = PXk = -k^λ = 1/2, λ >
0. Hence, show that if the WLLN holds, the central limit theorem holds. r.v.'s Xn के अनुक्रम के साथ PXk = k^λ = PXk = -k^λ = 1/2, λ > 0, इसलिए यदि WLLN होल्ड करता है, तो सिद्ध करें कि केन्द्रीय सीमा कानून/central limit होल्ड करता है।\n(b) A drunk performs a random walk over positions 0, ±1, ±2,... as follows. He starts at
0. He takes successive one-unit steps, going to the right with probability p and to the left with probability (1-p). His steps are independent. Let X denote his position after n steps. Find the distribution of (X + n)/2. Find E
(X) . एक शराबी अनियंत्रित चाल चलता है जिसकी स्थिति 0,±1, ±2,... Y.\n
(c) The following data represent the yields of maize in q/ha recorded from an experiment :\n16.4, 19.2, 24.5, 15.4, 17.3, 23.6, 22.7, 20.9, 18.2\nTest whether the median (M) is 20 q/ha. जाँच कीजिए कि माध्यिका (M) 20 q/ha है।\n(d) If the values of the units in the population consists of a linear trend such that Yi = a + bi (i = 1,2,...,N), then prove that Var(\barYst) ≤ Var(\barYsys)
() (c) The following data represent the yields of maize in q/ha recorded from an experiment : 16.4, 19.2, 24.5, 15.4, 17.3, 23.6, 22.7, 20.9, 18.2. Test whether the median (M) is 20 q/ha. जाँच कीजिए कि माध्यिका (M) 20 q/ha है.
() (d) If the values of the units in the population consists of a linear trend such that Yi = a + bi (i = 1,2,...,N), then prove that Var(\barYst)
