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यदि वास्तविक रेखा पर f(x) कोई प्रायिकता घनत्व है तो निम्न में से कौन-सा वैध प्रायिकता घनत्व नहीं है?

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  1. f(x + 1)
  2. f(2x)
  3. 2f(2x − 1)
  4. 3x2f(x3)

Answer (Detailed Solution Below)

Option 2 : f(2x)
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अवधारणा:

प्रायिकता घनत्व फलन f(x) = 1 को संतुष्ट करता है।

व्याख्या:

दिया गया है कि f(x) एक प्रायिकता घनत्व फलन है, इसलिए

= 1.....(i)

(1):

= (मान लीजिए x + 1 = y ⇒ dx = dy)

= 1 (समीकरण (i) का उपयोग करके)

इसलिए f(x + 1) एक प्रायिकता घनत्व फलन है।

विकल्प (1) गलत है।

(2):

= (मान लीजिए 2x= y ⇒ dx = dy)

= (समीकरण (i) का उपयोग करके)

इसलिए f(2x) एक मान्य प्रायिकता घनत्व फलन नहीं है।

विकल्प (2) सही है।

(3):

= (मान लीजिए 2x - 1 = y ⇒ 2dx = dy)

= 1 (समीकरण (i) का उपयोग करके)

इसलिए 2f(2x - 1) एक प्रायिकता घनत्व फलन है।

विकल्प (3) गलत है।

(4):

= (मान लीजिए x3 = y ⇒ 3x2dx = dy)

= 1 (समीकरण (i) का उपयोग करके)

इसलिए 3x2f(x3) एक प्रायिकता घनत्व फलन है।

विकल्प (4) गलत है।

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