एक यादृच्छिक चर x के लिए, सभी क्रम के केंद्रीय क्षण (\(\mu \)i) मौजूद हैं। क्षण (\(\mu^2_2{_j}{_+}{_1}\)) के (2j + 1)वें का वर्ग क्या है?

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SSC CGL JSO Tier-II, 2018 (Statistics) Official Paper-III (Held On: 14 Sept, 2019)
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  1. \(\mu_2{_j}\mu_2{_j}{_+}{_2}\) से अधिक
  2. \(\mu_2{_j}\mu_2{_j}{_+}{_2}\) से कम या इसके बराबर
  3. \(\mu_2{_j}\mu_2{_j}{_+}{_2}\) से अधिक या इसके बराबर
  4. \(\mu_2{_j}\mu_2{_j}{_+}{_2}\) से कम

Answer (Detailed Solution Below)

Option 2 : \(\mu_2{_j}\mu_2{_j}{_+}{_2}\) से कम या इसके बराबर
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Detailed Solution

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व्याख्या

एक यादृच्छिक चर के लिए, सभी क्रम के केंद्रीय क्षण (μ) मौजूद हैं

क्षण (\(\mu^2_2{_j}{_+}{_1}\)) के (2j + 1)वें का वर्ग

मान लीजिये कि j = 1 है तो (2j + 1)वां क्षण निम्न रूप से दिया गया है

23) = [(∑(x – x̅)3/N]2

⇒ (∑x3/N)2    ------(i)

यदि j = 1 तो μ2 = ∑x2/N

⇒  μ2j + 2 यदि j = 1 तो

⇒ μ4 = ∑x4/N

⇒ μ2 × μ2j + 2 = ∑x2/N × ∑x4/N    ------(ii)

दोनों समीकरणों की तुलना करने पर

(∑x3)2/N2 = (∑x2 × ∑x4)/N2  

⇒ (∑x3)2 ≤ (∑x2 × ∑x4)

 (μ2j + 1)2 ≤ μ2j × μ2j + 2

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