CDF of transformed variable

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

1. Given that \(Y = \frac { 1 } { X }\), find the (cumulative) distribution function of \(Y\), and deduce that \(Y\) and \(X\) have identical distributions.
2. Find \(\mathrm { E } ( X + 1 )\) and deduce the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).

7 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 \\ 1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$

1. Find the (cumulative) distribution function, \(\mathrm { G } ( y )\), of the random variable \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
2. Hence show that the probability density function of \(Y\) is given by $$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
3. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).

4 The continuous random variable \(V\) has (cumulative) distribution function given by $$\mathrm { F } ( v ) = \begin{cases} 0 & v < 1 \\ 1 - \frac { 8 } { ( 1 + v ) ^ { 3 } } & v \geqslant 1 \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { 1 + V }\).

1. Show that the (cumulative) distribution function of \(Y\) is \(8 y ^ { 3 }\), over an interval to be stated, and find the probability density function of \(Y\).
2. Find \(\mathrm { E } \left( \frac { 1 } { Y ^ { 2 } } \right)\).

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \sqrt { x } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { \sqrt { X } }\).

1. Find the (cumulative) distribution function of \(Y\), and hence show that its probability density function is given by $$\mathrm { g } ( y ) = \frac { 3 } { y ^ { 4 } }$$ for a set of values of \(y\) to be stated.
2. Find the value of \(\mathrm { E } \left( Y ^ { 2 } \right)\).

9 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find the distribution function of \(Y\). Given that \(a = 4\), find the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = 0.25\).

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. \(\mathrm { E } ( Y )\).

9 The continuous random variable \(T\) has cumulative distribution function \(F ( t ) = \begin{cases} 0 & t < 0 , \\ 1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}\)

1. Find the cumulative distribution function of \(2 T\).
2. Show that, for constant \(k , \mathrm { E } \left( \mathrm { e } ^ { k t } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
3. \(\quad T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\). \section*{END OF QUESTION PAPER}

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}

1. A game uses two turntables, one red and one yellow. Each turntable has a point marked on the circumference that is lined up with an arrow at the start of the game. Jim spins both turntables and measures the distance, in metres, each point is from the arrow, around the circumference in an anticlockwise direction when the turntables stop spinning.

The continuous random variable \(Y\) represents the distance, in metres, the point is from the arrow for the yellow turntable. The cumulative distribution function of \(Y\) is given by \(\mathrm { F } ( y )\) where $$F ( y ) = \left\{ \begin{array} { c r } 0 & y < 0 \\ 1 - \left( \alpha + \beta y ^ { 2 } \right) & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{array} \right.$$

1. Explain why (i) \(\alpha = 1\) $$\text { (ii) } \beta = - \frac { 1 } { 25 }$$
2. Find the probability density function of \(Y\) The continuous random variable \(R\) represents the distance, in metres, the point is from the arrow for the red turntable. The distribution of \(R\) is modelled by a continuous uniform distribution over the interval \([ d , 3 d ]\) Given that \(\mathrm { P } \left( R > \frac { 11 } { 5 } \right) = \mathrm { P } \left( Y > \frac { 5 } { 3 } \right)\)
3. find the value of \(d\) In the game each turntable is spun 3 times. The distance between the point and the arrow is determined for each spin. To win a prize, at least 5 of the distances the point is from the arrow when a turntable is spun must be less than \(\frac { 11 } { 5 } \mathrm {~m}\) Jo plays the game once.
4. Calculate the probability of Jo winning a prize.

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]

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

1. The random variable \(Y\) is defined by \(Y = \frac{1}{X^2}\). Find the cumulative distribution function of \(Y\). [5]
2. Show that E\((Y)\) is not defined. [4]