Geometric/graphical PDF with k

3
\includegraphics[max width=\textwidth, alt={}, center]{9fca48da-82c3-4ce1-9e0c-93eb9a920f9d-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.

1. Show that \(p = 2\).
2. Find \(\mathrm { E } ( X )\).

7

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{593c1ece-82a2-4dcd-8041-f39c98adf631-12_357_738_267_731} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 4 only. Between these two values the graph is a straight line.
   1. Show that \(\mathrm { f } ( x ) = k x\) for \(0 \leqslant x \leqslant 4\), where \(k\) is a constant to be determined.
   2. Hence, or otherwise, find \(\mathrm { E } ( X )\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{593c1ece-82a2-4dcd-8041-f39c98adf631-13_383_752_269_731} The diagram shows the graph of the probability density function, g , of a random variable \(W\) which takes values between 0 and \(a\) only, where \(a > 0\). Between these two values the graph is a straight line. Given that the median of \(W\) is 1 , find the value of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6
\includegraphics[max width=\textwidth, alt={}, center]{5dfbc896-528c-40a6-a296-8a0aae90add4-10_451_469_255_799} The diagram shows the graph of the probability density function, f , of a random variable \(X\). The graph is a quarter circle entirely in the first quadrant with centre \(( 0,0 )\) and radius \(a\), where \(a\) is a positive constant. Elsewhere \(\mathrm { f } ( x ) = 0\).

1. Show that \(a = \frac { 2 } { \sqrt { \pi } }\).
2. Show that \(\mathrm { f } ( x ) = \sqrt { \frac { 4 } { \pi } - x ^ { 2 } }\).
3. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 3 \sqrt { \pi ^ { 3 } } }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{image-not-found} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(k\).
2. Using this value of \(k\), find \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant k\) and hence find \(\mathrm { E } ( X )\).
3. Find the value of \(p\) such that \(\mathrm { P } ( p < X < 1 ) = 0.25\).

1
\includegraphics[max width=\textwidth, alt={}, center]{cfffe79d-91c9-48b8-a3e6-887d7891441d-2_478_691_260_724} The random variable \(X\) has probability density function, f , as shown in the diagram, where \(a\) is a constant. Find the value of \(a\) and hence show that \(\mathrm { E } ( X ) = 0.943\) correct to 3 significant figures. [5]

3
\includegraphics[max width=\textwidth, alt={}, center]{189bcf7b-279f-457b-8232-ace7f0c9797f-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.

1. Show that \(p = 2\).
2. Find \(\mathrm { E } ( X )\).

3
\includegraphics[max width=\textwidth, alt={}, center]{ec7cab36-683b-4022-9cac-fb3b4e64778a-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.

1. Show that \(c = \frac { 2 } { 3 }\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Calculate \(\mathrm { E } ( X )\).

3
\includegraphics[max width=\textwidth, alt={}, center]{4a5f9f7e-b045-4c6f-8bda-6c4067668da2-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.

1. Show that \(c = \frac { 2 } { 3 }\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Calculate \(\mathrm { E } ( X )\).

2
\includegraphics[max width=\textwidth, alt={}, center]{2fefee17-50bb-4375-80f6-7e4bc2606492-04_405_789_260_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(k\).
2. Using this value of \(k\), find \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant k\) and hence find \(\mathrm { E } ( X )\).
3. Find the value of \(p\) such that \(\mathrm { P } ( p < X < 1 ) = 0.25\).

2
\includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(c\).
2. Find the value of \(a\) such that \(\mathrm { P } ( a < X < 1 ) = 0.1\).
3. Find \(\mathrm { E } ( X )\).

7 The continuous random variable \(X\) has the probability density function shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}

1. Find the value of the constant \(k\).
2. Write down the mean of \(X\), and use integration to find the variance of \(X\).
3. Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
4. The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance. \section*{[Question 8 is printed overleaf.]}

2 The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only with probability density function f . The graph of \(y = \mathrm { f } ( x )\) consists of the two line segments shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{4a6d94a2-66e1-449a-ac0e-1fbada74bb3b-2_524_1287_950_429}

1. Show that \(a = \frac { 2 } { 3 }\).
2. Find the equations of the two line segments.
3. Hence write down the probability density function of \(X\).
4. Find \(\mathrm { E } ( X )\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 ,
\frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 ,
x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 ,
\frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 ,
x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 5\) only. For \(0 \leqslant x \leqslant 5\) the graph of its probability density function f consists of two straight line segments, as shown in the diagram. Find \(k\) and show that f is given by $$f ( x ) = \begin{cases} \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 2
\frac { 1 } { 4 } & 2 < x \leqslant 5
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\).

1. Find the probability density function of \(Y\).
2. Show that \(\mathrm { E } ( Y ) = 10.25\).
3. Show that the median of \(Y\) is the square of the median of \(X\).

4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 ,
\mathrm { f } ( x ) = k x & 1 \leq x \leq 4 ,
\mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$

1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}