Question 1
A CRV has PDF \(f(x)=egin{cases}ax & 0\leq x\leq2\ b & 2
Given that \(f\) is continuous at \(x=2\) and is a valid PDF, find \(a\) and \(b\).
Continuity at \(x=2\)
\(2a=b\)
Integral equals 1
\(\displaystyle\int_0^2 ax\,dx+\displaystyle\int_2^4 b\,dx=[a frac{x^2}{2}]_0^2+2b=2a+2b=1\)
Substitute \(b=2a\)
\(2a+4a=1\Rightarrow a= frac{1}{6}\)
Answer
\(a=\dfrac{1}{6},\;b=\dfrac{1}{3}\)
Question 2
The CDF of \(X\) is \(F(x)=egin{cases}0&x<0\x^2&0\leq x\leq1\1&x>1\end{cases}\)
(a)Find the PDF \(f(x)\).
(b)Find \(E(X)\).
(c)Find the median.
(a)
Differentiate CDF
\(f(x)=2x\) for \(0\leq x\leq1\)
Answer
\(f(x)=2x,\;0\leq x\leq1\)
(b)
Integrate
\(\displaystyle\int_0^12x^2\,dx= frac{2}{3}\)
Answer
\(E(X)=\dfrac{2}{3}\)
(c)
Solve \(F(m)=0.5\)
\(m^2=0.5\Rightarrow m= frac{1}{\sqrt2}\)
Answer
\(m=\dfrac{\sqrt2}{2}pprox0.707\)
Question 3
For the PDF \(f(x)=\dfrac{3}{4}(1-x^2),\;-1\leq x\leq1\), find \( ext{Var}(X)\).
By symmetry \(E(X)=0\)
\( ext{Var}(X)=E(X^2)=\displaystyle\int_{-1}^1 frac{3}{4}x^2(1-x^2)\,dx= frac{3}{4}\displaystyle\int_{-1}^1(x^2-x^4)\,dx\)
Integrate
\( frac{3}{4}\cdot2\int_0^1(x^2-x^4)\,dx= frac{3}{2}[ frac{x^3}{3}- frac{x^5}{5}]_0^1= frac{3}{2}\cdot frac{2}{15}\)
Answer
\( ext{Var}(X)=\dfrac{1}{5}\)
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