5.03g Cdf of transformed variables

9 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 2 \\ a \mathrm { e } ^ { - ( x - 2 ) } & x \geqslant 2 \end{cases}$$ where \(a\) is a constant. Show that \(a = 1\). Find the distribution function of \(X\) and hence find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find

1. the probability density function of \(Y\),
2. \(\mathrm { P } ( Y > 10 )\).

8 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 2 \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\).
3. Find the probability density function of \(Y\).

9 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } \left( 3 - \frac { 1 } { \sqrt { } x } \right) & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = \sqrt { } X\).

1. Show that the probability density function of \(Y\) is given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 10 } ( 3 y - 1 ) & 1 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the mean value of \(Y\).

6 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 1 \\ \frac { 1 } { 2 } & 1 \leqslant x \leqslant 3 \\ 0 & x > 3 \end{cases}$$ Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find

1. the probability density function of \(Y\),
2. the expected value and variance of \(Y\).

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 2 } { 15 } x & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Show that the distribution function G of \(Y\) is given by $$\mathrm { G } ( y ) = \begin{cases} 0 & y < 1 \\ \frac { 1 } { 15 } \left( y ^ { \frac { 2 } { 3 } } - 1 \right) & 1 \leqslant y \leqslant 64 \\ 1 & y > 64 \end{cases}$$ Find

1. the median value of \(Y\),
2. \(\mathrm { E } ( Y )\).

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 \(X\).

10 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find the distribution function of \(Y\). Sketch the graph of the probability density function of \(Y\). Find the probability that \(Y\) lies between its median value and its mean value.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\). Show that \(Y\) has probability density function given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$ Find

1. the median value of \(Y\),
2. the expected value of \(Y\).

7 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find
2. the probability density function of \(Y\),
3. the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = \frac { 7 } { 12 }\).

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).

1. Show that \(Y\) has probability density function given by $$g ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the median value of \(Y\).
3. Find the expected value of \(Y\).

6 The continuous random variable \(X\) has a uniform distribution on the interval \([ - \pi , \pi ]\).
The random variable \(Y\) is defined by \(Y = \sin X\).
Determine the cumulative distribution function of \(Y\).

9 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 16 } x ^ { 2 } & 0 \leq x \leq 4 \\ 1 & x > 4 \end{array} \right.$$

1. The random variable \(Y\) is defined by \(Y = \frac { 1 } { X ^ { 2 } }\). Find the cumulative distribution function of \(Y\).
2. Show that \(\mathrm { E } ( Y )\) is not defined. \section*{END OF QUESTION PAPER}

7 The sides of a square are each of length \(L \mathrm {~cm}\) and its area is \(A \mathrm {~cm} ^ { 2 }\) Given that \(A\) is uniformly distributed on the interval [10,30]

1. find \(\mathrm { P } ( L \geqslant 4.5 )\)
2. find \(\operatorname { Var } ( L )\)
   \includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}

6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).

1. Find \(\mathrm { P } ( A > 3 )\).
2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
3. Find the probability density function of \(Y\).
4. Sketch the probability density function of \(Y\).
5. Write down the mode of \(Y\).
6. Find \(\mathrm { E } ( Y )\).
7. Find \(\mathrm { P } ( Y > 3 )\).

1. The random variable \(X\) has the continuous uniform distribution over the interval [0.5, 2.5]

Talia selects a number, \(T\), at random from the distribution of \(X\)

1. Find \(\mathrm { P } ( T < 1 )\) Malik takes Talia's number, \(T\), and calculates his number, \(M\), where \(M = \frac { 1 } { T ^ { 2 } }\)
2. Find the probability that both \(T\) and \(M\) are less than 2.25 Raja and Greta play a game many times.
   Each time they play they use a number, \(R\), randomly selected from the distribution of \(X\) Raja's score is \(R\) Greta's score is \(G\), where \(G = \frac { 2 } { R ^ { 2 } }\)
3. Determine, giving a reason, who you would expect to have the higher total score.

1. The continuous random variable \(X\) has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 1 \\ 1.5 x - 0.25 x ^ { 2 } - 1.25 & 1 \leqslant x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$

1. Find the exact value of the median of \(X\)
2. Find \(\mathrm { P } ( X < 1.6 \mid X > 1.2 )\) The random variable \(Y = \frac { 1 } { X }\)
3. Specify fully the cumulative distribution function of \(Y\)
4. Hence or otherwise find the mode of \(Y\)

8 A continuous random variable \(X\) has probability density function given by the following function, where \(a\) is a constant. \(\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise. } \end{array} \right\}\) The expected value of \(X\) is 4 .

1. Show that \(a = 6\). Five independent observations of \(X\) are obtained, and the largest of them is denoted by \(M\).
2. Find the cumulative distribution function of \(M\). \section*{OCR} Oxford Cambridge and RSA

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5 \\ 0 & \text { otherwise } \end{cases}$$ 6

1. Find \(\mathrm { P } ( X > 2 )\) [0pt] [2 marks]
   6
2. Find the upper quartile of \(X\) Give your answer to two decimal places.
   6
3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.

4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 8 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).

The lifetime, \(X\) days, of a particular insect is such that \(\log_{10} X\) has a normal distribution with mean \(1.5\) and standard deviation \(0.2\). Find the median lifetime. [3] Find also P\((X \geqslant 50)\). [2]

The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} \frac{1}{4}(x - 1) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise.} \end{cases}$$

1. Find the distribution function of \(X\). [3]

The random variable \(Y\) is defined by \(Y = (X - 1)^3\).

1. Find the probability density function of \(Y\). [4]
2. Find the median value of \(Y\). [3]

The continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{1}{4}(x - 1) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the distribution function of \(X\). [3]
2. The random variable \(Y\) is defined by \(Y = (X - 1)^3\). Find the probability density function of \(Y\). [4]
3. Find the median value of \(Y\). [3]

The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\). [2]
2. State the value of E\((X)\). [1]
3. Find the interquartile range of \(X\). [4]

The variable \(Y\) is related to \(X\) by \(Y = 2^X\).

1. Find the probability density function of \(Y\). [5]

A continuous random variable \(X\) has cumulative distribution function F given by $$\mathrm{F}(x) = \begin{cases} 0 & x < -1, \\ \frac{1}{4}(x^3 + 1) & -1 \leqslant x \leqslant 1, \\ 1 & x > 1. \end{cases}$$ Find \(\mathrm{P}\left(X \geqslant \frac{3}{4}\right)\), and state what can be deduced about the upper quartile of \(X\). [3] Obtain the cumulative distribution function of \(Y\), where \(Y = X^2\). [5]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{2} & 1 \leq x \leq 3, \\ 0 & \text{otherwise.} \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^3\). Find the distribution function of \(Y\). [5] Sketch the graph of the probability density function of \(Y\). [3] Find the probability that \(Y\) lies between its median value and its mean value. [4]