Question

For the random variable X having PDF f(x) = 4x³; 0 < x < 1, the interquartile range is:
1. \(\left( \dfrac{3}{4} \right)^{\dfrac{1}{4}} - \left( \dfrac{1}{4} \right)^{\dfrac{1}{4}}\)
2. \(\left( \dfrac{3}{4} - \dfrac{1}{4} \right)^4\)
3. \(\left( \dfrac{3}{4} - \dfrac{1}{4} \right)^\dfrac{1}{4}\)
4. \(\left( \dfrac{1}{4} \right)^\dfrac{1}{4} - \left( \dfrac{3}{4} \right)^\dfrac{1}{4}\)

Answer 1

Correct Answer - Option 1 : \(\left( \dfrac{3}{4} \right)^{\dfrac{1}{4}} - \left( \dfrac{1}{4} \right)^{\dfrac{1}{4}}\)

Given

F(x) = 4x³

Formula

For interquartile range = Q₃ – Q₁

Q₃ = 75/100

Q₁ = 25/100

Calculation

\(\mathop \smallint \nolimits_o^{Q3} \)4x³dx = 75/100

⇒ ¾ = 4 (X⁴/4)^Q3₀

⇒ ¾ = (Q⁴₃ – 0))

⇒ ¾ = Q⁴₃

∴ Q₃ = (3/4)^1/4

For Quartile (Q₁),

⇒ 25/100 = \(\mathop \smallint \nolimits_o^{Q1} \)4x³dx

⇒ ¼ = 4(x⁴/4)^Q1₀

⇒ ¼ = (Q₁⁴ – 0)

⇒ ¼ = Q₁⁴

∴ Q₁ = (1/4)^1/4

∴ Interquartile range Q₃ – Q₁ IS (3/4)^1/4  - (1/4)^1/4