निरंतर यादृच्छिक चर X की अन्तःचतुर्थक परास जिसका प्रायिकता घनत्व फलन f(x) = e-x; x ≥ 0 है:

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SSC CGL Tier-II ( JSO ) 2019 Official Paper ( Held On : 17 Nov 2020 )
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  1. \(\rm In\dfrac{1}{2}\)
  2. In 3
  3. In 2
  4. \(\rm In\dfrac{1}{3}\)

Answer (Detailed Solution Below)

Option 2 : In 3
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दिया है

F(x) = e-x

गणना

\(Q_3=\displaystyle\int^{Q_3}_0e^{-x}dx=\dfrac{75}{100}\) के लिये

⇒ ¾ = (-e-x)Q30

⇒ ¾ = (-1 – e-Q3)

⇒ ¼ = e-Q3

⇒Q3 = In 4

\(Q_1=\displaystyle\int^{Q_1}_0e^{-x}dx=\dfrac{25}{100}\) के लिये

⇒ (-e-xdx)Q10 = 1/4

⇒ (- 1 – e-Q1) = ¼

⇒ Log Q1 = ¾

Q1 = In 4/3

⇒ अन्तःचतुर्थक परास = Q3 – Q1

⇒ In 4 – In 4/3

⇒ In  4 –(In 4 – In 3)

∴ अन्तःचतुर्थक परास In 3 है।

प्रायिकता घनत्व फलन = \(\displaystyle\int_0^\infty f(x)\ dx=1\)

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