Calculate probability P(X in interval)

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

1. Find \(\mathrm { P } ( X > 1.5 )\).
2. Find the mean of \(X\).
3. Find the median of \(X\).

6 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 3 ( 1 - x ) ^ { 2 } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ Find

1. \(\mathrm { P } ( X > 0.5 )\),
2. the mean and variance of \(X\).

3

1. The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by $$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2
   0 & \text { otherwise } \end{cases}$$ Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{f9436a68-ec88-4feb-9c06-fc29fe53d1fe-2_405_793_1471_715} The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
   1. Find the value of \(a\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).

3

1. The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by $$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2
   0 & \text { otherwise } \end{cases}$$ Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{c08d3228-430e-4158-9362-1655deb1feb7-2_405_791_1471_715} The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
   1. Find the value of \(a\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).

5 Th co in rach \& riab e \(X \mathbf { h }\) s prb b lity e \(\mathbf { s }\) ityf \(\mathbf { n }\) tiff \(\dot { \mathrm { g } }\) \& $$f ( x ) = \begin{cases} 0 & x < 0
\frac { 6 } { 5 } x & 0 \leqslant x \leqslant 1
\frac { 6 } { 5 } x ^ { - 4 } & x > 1 \end{cases}$$

1. FidP \(( X > 1\).
2. Fid b med arm le \(6 X\).
3. Gie it \(\mathbf { h } \mathrm { t } \mathrm { E } ( X ) =\), fif id \(\mathbf { b }\) riance \(6 X\).
4. Fide \(( \sqrt { X } )\).

1. The continuous random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 48 } \left( x ^ { 2 } - 8 x + c \right) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Show that \(c = 31\)
2. Find \(\mathrm { P } ( 2 < X < 3 )\)
3. State whether the lower quartile of \(X\) is less than 3, equal to 3 or greater than 3 Give a reason for your answer. Kei does the following to work out the mode of \(X\) $$\begin{aligned} f ^ { \prime } ( x ) & = \frac { 1 } { 48 } ( 2 x - 8 )
   0 & = \frac { 1 } { 48 } ( 2 x - 8 )
   x & = 4 \end{aligned}$$ Hence the mode of \(X\) is 4 Kei's answer for the mode is incorrect.
4. Explain why Kei's method does not give the correct value for the mode.
5. Find the mode of \(X\) Give a reason for your answer.

7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \begin{cases} k \left( x ^ { 2 } + 2 x + 1 \right) & - 1 \leq x \leq 0
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.

1. Show that \(k = 3\). Find
2. \(\mathrm { E } ( X )\),
3. the cumulative distribution function \(\mathrm { F } ( x )\),
4. \(\mathrm { P } ( - 0.3 < X < 0.3 )\). END

3 At a factory, flour is packed into bags. A model for the mass in grams of flour packed into each bag is \(1500 + X\), where \(X\) is a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x ( 6 - x ) & 0 \leq x \leq 6
0 & \text { elsewhere, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 36 }\).
2. Find the probability that a randomly selected bag of flour contains 1505 grams of flour or more.
3. Find
   • the mean of \(X\),
   • the standard deviation of \(X\).

11 The length of time in minutes for which a particular geyser erupts is modelled by the continuous random variable \(T\) with cumulative distribution function given by
\(\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 2 ,
k \left( 8 t ^ { 2 } - t ^ { 3 } - 24 \right) & 2 < t < 4 ,
1 & t \geqslant 4 , \end{cases}\)
where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 40 }\).
2. Find the probability that a randomly selected eruption time lies between 2.5 and 3.5 minutes.
3. Show that the median \(m\) of the distribution satisfies the equation \(m ^ { 3 } - 8 m ^ { 2 } + 44 = 0\).
4. Verify that the median eruption time is 2.95 minutes, correct to 2 decimal places. The mean and standard deviation of \(T\) are denoted by \(\mu\) and \(\sigma\) respectively.
5. Find \(\mathrm { P } ( \mu - \sigma < T < \mu + \sigma )\).
6. Sketch the graph of the probability density function of \(T\).
7. A Normally distributed random variable \(X\) has the same mean and standard deviation as \(T\). By considering the shape of the Normal distribution, and without doing any calculations, explain whether \(\mathrm { P } ( \mu - \sigma < X < \mu + \sigma )\) will be greater than, equal to or less than the probability that you calculated in part (e).

2 The continuous random variable \(Y\) has cumulative distribution function defined by $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { y ^ { 2 } } { 36 } & 0 \leq y \leq 6
1 & y > 6 \end{array} \right.$$ Find the value of \(\mathrm { P } ( Y > 4 )\)
Circle your answer.
\(\frac { 4 } { 9 }\)
\(\frac { 5 } { 9 }\)
\(\frac { 16 } { 27 }\)
\(\frac { 11 } { 27 }\)

4 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - c
\frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c
1 & x > 3 c \end{array} \right.$$ where \(c\) is a positive constant.

1. Determine \(\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)\).
2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c
   0 & \text { otherwise } \end{array} \right.$$
3. Hence, or otherwise, find expressions, in terms of \(c\), for:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).

1 Let \(X\) be a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( 2 - x ) & 0 \leq x \leq 2
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X = 1 )\)
Circle your answer.
0
\(\frac { 1 } { 2 }\)
\(\frac { 3 } { 4 }\)
\(\frac { 27 } { 32 }\)

2 The random variable \(X\) has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 }
\frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 }
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X < 1 )\) Circle your answer.
[0pt] [1 mark]
\(\frac { 1 } { 8 }\)
\(\frac { 3 } { 8 }\)
\(\frac { 5 } { 8 }\)
\(\frac { 7 } { 8 }\)
\includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}