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Continuous Random Variables I
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
The continuous random variable \(X\) has PDF \(f(x)=3x^2\) for \(0\leq x\leq1\) and 0 otherwise.
(a)Verify that \(f(x)\) is a valid PDF.
(b)Find \(P(0.5\leq X\leq0.8)\).
(c)Find \(E(X)\) and \( ext{Var}(X)\).
(d)Find the median.
(a)
Integrate over all \(x\)
\(\displaystyle\int_0^13x^2\,dx=[x^3]_0^1=1\) ✓
(b)
Integrate
\([x^3]_{0.5}^{0.8}=0.512-0.125\)
Answer
\(0.387\)
(c)
\(E(X)\)
\(\displaystyle\int_0^1 3x^3\,dx= frac{3}{4}\)
\(E(X^2)\)
\(\displaystyle\int_0^13x^4\,dx= frac{3}{5}\)
Answer
\(E(X)= frac{3}{4}\); \( ext{Var}(X)= frac{3}{5}- frac{9}{16}= frac{3}{80}\)
(d)
Solve \(\displaystyle\int_0^m 3x^2\,dx= frac{1}{2}\)
\(m^3= frac{1}{2}\Rightarrow m= frac{1}{\sqrt[3]{2}}\)
Answer
\(m=2^{-1/3}pprox0.794\)
Question 2
A CRV \(X\) has PDF \(f(x)=k(4-x^2)\) for \(-2\leq x\leq2\), and 0 otherwise.
(a)Find \(k\).
(b)Find \(P(0\leq X\leq1)\).
(c)State the mode and justify why \(E(X)=0\).
(a)
Integrate
\(k\displaystyle\int_{-2}^2(4-x^2)\,dx=k[4x- frac{x^3}{3}]_{-2}^2=k\cdot frac{32}{3}=1\Rightarrow k= frac{3}{32}\)
Answer
\(k=\dfrac{3}{32}\)
(b)
Integrate with \(k\)
\( frac{3}{32}[4x- frac{x^3}{3}]_0^1= frac{3}{32} imes frac{11}{3}= frac{11}{32}\)
Answer
\(\dfrac{11}{32}\)
(c)
Mode at \(f'(x)=0\)
\(-2x=0\Rightarrow x=0\). Mode \(=0\). \(E(X)=0\) by symmetry (odd integrand on symmetric interval).
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