Verify algebraic PDF formula

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\includegraphics[max width=\textwidth, alt={}, center]{10cf346f-dee2-4223-8caa-2a49f1eaa99f-10_547_880_260_621} A random variable \(X\) has probability density function f , where the graph of \(y = \mathrm { f } ( x )\) is a semicircle with centre \(( 0,0 )\) and radius \(\sqrt { \frac { 2 } { \pi } }\), entirely above the \(x\)-axis. Elsewhere \(\mathrm { f } ( x ) = 0\) (see diagram).

1. Verify that f can be a probability density function.
   \(A\) and \(B\) are the points where the line \(x = \sqrt { \frac { 1 } { \pi } }\) meets the \(x\)-axis and the semicircle respectively.
2. Show that angle \(A O B\) is \(\frac { 1 } { 4 } \pi\) radians and hence find \(\mathrm { P } \left( X > \sqrt { \frac { 1 } { \pi } } \right)\).

5 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$

1. Show that this is a valid probability density function. [4]
2. Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
3. Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
4. Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
5. A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]

5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e }
0 & \text { otherwise } \end{cases}$$

1. Verify that this is a valid probability density function.
2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1
   y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e }
   1 & \text { otherwise } \end{cases}$$
3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). For \(1 \leqslant x \leqslant 2 , \mathrm { f } ( x )\) is represented by a curve with equation \(\mathrm { f } ( x ) = k \left( \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + a x + 1 \right)\) where \(k\) and \(a\) are constants. For all other values of \(x , \mathrm { f } ( x ) = 0\)

1. Use algebraic integration to show that \(k ( 12 a - 33 ) = 8\) Given that \(a = 5\)
2. calculate the mode of \(X\).

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7. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 6 x } { 175 } & 0 \leq x < 5
\mathrm { f } ( x ) = \frac { 6 x ( 10 - x ) } { 875 } & 5 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$

1. Verify that f is a probability density function.
2. Write down the probability that \(X < 1\).
3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains.
4. Find the probability that \(2 < X < 7\).

7 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { f } ( x )\) satisfies the conditions for a probability density function.
2. Find the value of \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 15 } { 16 }\).