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Continuous Random Variables I
IB Mathematics AA · HL · Shadow Worksheet
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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 \(\text{Var}(X)\).
(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)
\(E(X)\)
\(\displaystyle\int_0^1 3x^3\,dx=\tfrac{3}{4}\)
\(E(X^2)\)
\(\displaystyle\int_0^13x^4\,dx=\tfrac{3}{5}\)
Answer
\(E(X)=\tfrac{3}{4}\); \(\text{Var}(X)=\tfrac{3}{5}-\tfrac{9}{16}=\tfrac{3}{80}\)
(d)
Solve \(\displaystyle\int_0^m 3x^2\,dx=\tfrac{1}{2}\)
\(m^3=\tfrac{1}{2}\Rightarrow m=\tfrac{1}{\sqrt[3]{2}}\)
Answer
\(m=2^{-1/3}\approx0.794\)
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
\(k\displaystyle\int_{-2}^2(4-x^2)\,dx=k[4x-\tfrac{x^3}{3}]_{-2}^2=k\cdot\tfrac{32}{3}=1\Rightarrow k=\tfrac{3}{32}\)
Answer
\(k=\dfrac{3}{32}\)
(b)
Integrate with \(k\)
\(\tfrac{3}{32}[4x-\tfrac{x^3}{3}]_0^1=\tfrac{3}{32}\times\tfrac{11}{3}=\tfrac{11}{32}\)
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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