Find constant k in PDF

1 \includegraphics[max width=\textwidth, alt={}, center]{29ab8740-9fad-4596-b34f-c424b5464858-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.

1. State the value of \(k\).
2. Explain briefly what this graph tells you about the arrival times of trains.

1 \includegraphics[max width=\textwidth, alt={}, center]{f73b303b-eb56-4c35-aa3e-388e3a3e8acd-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.

1. State the value of \(k\).
2. Explain briefly what this graph tells you about the arrival times of trains.

7 Two continuous random variables \(S\) and \(T\) have probability density functions \(\mathrm { f } _ { S }\) and \(\mathrm { f } _ { T }\) given respectively by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Sketch on the same axes the graphs of \(y = \mathrm { f } _ { S } ( x )\) and \(y = \mathrm { f } _ { T } ( x )\).
2. Find the value of \(a\).
3. Find \(\mathrm { E } ( S )\).
4. A student gave the following description of the distribution of \(T\) : "The probability that \(T\) occurs is constant". Give an improved description, in everyday terms.

4

1. A random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(k = \frac { 1 } { b - a }\).
   2. Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
2. The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
   1. Write down the value of the mean, \(\mu\), of \(X\).
   2. Evaluate the standard deviation, \(\sigma\), of \(X\).
   3. Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).

2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\ \mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
2. Write down the value of \(\mathrm { P } ( X < 8 )\).

The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$

1. 1. Sketch the graph of f. [2 marks]
   2. Hence show that \(k = \frac{1}{2}\). [2 marks]
2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
3. 1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
   2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]

A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$

1. State the value of \(k\). [1 mark]
2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
3. 1. State the value for E(\(X\)).
   2. Use integration to find the value for E(\(X^2\)).
   3. Hence find the value for the standard deviation of \(X\).
   [5 marks]