Calculate probabilities from CDF

7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of \(X\), find the probability that 2 lies between \(X\) and \(4 X\). Find also the expected value of the width of the interval ( \(X , 4 X\) ).

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$

1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
2. Use calculus to find (i) \(\mathrm { E } ( X )\)
   (ii) \(\operatorname { Var } ( X )\).

1. A continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 1 } { 16 } ( x - 1 ) ^ { 2 } & 1 \leqslant x \leqslant 5
1 & x > 5 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Find \(\mathrm { P } ( X > 3 \mid 2 < X < 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)

3. Figure 1 shows an accurate graph of the cumulative distribution function, \(\mathrm { F } ( x )\), for the continuous random variable \(X\) \begin{figure}[h] \end{figure}

1. Find \(\mathrm { P } ( 3 < X < 7 )\) The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4
   b & 4 \leqslant x < 6
   c & 6 \leqslant x \leqslant 8
   0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants.
2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
3. Find \(\mathrm { E } ( X )\)

2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable \(W\) with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0
\frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4
1 & w > 4 \end{array} \right.$$

1. Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
2. Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given \(\mathrm { E } ( W ) = 1.6\)
3. use algebraic integration to find \(\operatorname { Var } ( W )\)
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6. The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0
\frac { 1 } { 135 } \left( 54 t + 9 t ^ { 2 } - 4 t ^ { 3 } \right) , & 0 \leq t \leq 3
1 , & t > 3 \end{cases}$$

1. Find the probability that a patient waits for more than 20 minutes.
2. Show that the median waiting time is between 11 and 12 minutes.
3. Define fully the probability density function \(\mathrm { f } ( t )\) of \(T\).
4. Find the modal waiting time in minutes.
5. Give one reason why this model may need to be refined.

2 The cumulative distribution function of the continuous random variable, \(Y\), is given below. $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { y ^ { 3 } - y ^ { 2 } } { 4 } & 1 \leq y \leq 2
1 & y > 2 \end{array} \right.$$

1. Find \(\mathrm { P } ( Y \leq 1.5 )\)
2. Verify that the median of \(Y\) lies between 1.6 and 1.7.
3. Find the probability density function of \(Y\).

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 2
1.25 - \frac { 2.5 } { x } & 2 \leqslant x \leqslant 10
1 & x > 10 \end{array} \right.$$

1. Find \(\mathrm { P } ( \{ X < 5 \} \cup \{ X > 8 \} )\)
2. Find the median of \(X\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. 1. Sketch the probability density function of \(X\).
   2. Describe the skewness of the distribution of \(X\).