Question
Download Solution PDFयदि प्वासों वितरण के बाद एक यादृच्छिक चर X, के लिए प्रत्याशा λ है, तो निम्नलिखित में से कौन सा कथन सत्य है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFk = 0, 1, 2, ... के लिए \(P(X = k) = λ^k e^{(-λ)} {1 \over k!} \)
इसलिए, हम लिख सकते हैं: \(P(X = λ) = λ^λ e^{(-λ)} {1 \over λ!}\)और \(P(X = λ + 1) = λ^{(λ + 1)} e^{{(-λ)} \over (λ + 1)!}\)
इन दो प्रायिकताओं का अनुपात लेने पर, हमें प्राप्त होता है:
\({P(X = \lambda) \over P(X = \lambda + 1)} = {{{[\lambda^\lambda \times e^{(-\lambda)} {1 \over \lambda!}]} \over {[\lambda^{(\lambda + 1)} \times e^{(-\lambda)} {1 \over (\lambda + 1)!}]}}}\)
\(={{\lambda^\lambda \over {\lambda^{(\lambda+1)}}} \times {{(\lambda+1)!} \over λ!}}\)
\(= {\lambda +1 \over \lambda}\)
उपरोक्त समीकरण को पुनः व्यवस्थित करने पर, हमें दो प्रायिकताओं के बीच का संबंध प्राप्त होता है:
\(\rm P(X=\lambda)=\frac{\lambda +1}{\lambda}P(X=\lambda+1)\)
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