Find constant k in PDF

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\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.

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\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.

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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 \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).

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

1. 1. Sketch the graph of f.
   2. Hence show that \(k = \frac { 1 } { 2 }\).
2. Find the exact numerical values for the mean and the standard deviation of \(X\).
3. 1. Find \(\mathrm { P } \left( X \geqslant - \frac { 1 } { 4 } \right)\).
   2. Write down the value of \(\mathrm { P } \left( X \neq - \frac { 1 } { 4 } \right)\).
      
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-07_2484_1709_223_153}

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 \(\mathrm { P } ( X < 8 )\).