5.03e Find cdf: by integration

The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),

1. find the value of the constant \(k\). [3 marks]
2. Find P\((X < 0)\). [2 marks]
3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]

The discrete random variable \(X\) has the following probability distribution:

\(x\)	0	1	2	3	4	5
\(\text{P}(X = x)\)	0.11	0.17	0.2	0.13	\(p\)	\(p^2\)

1. Find the value of \(p\). [4 marks]
2. Find
   1. \(\text{P}(0 < X \leq 2)\),
   2. \(\text{P}(X \geq 3)\).
   [3 marks]
3. Find the mean and the variance of \(X\). [3 marks]
4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]

The random variable \(X\) has probability density function defined by $$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. State values for the median and the lower quartile of \(X\). [2 marks]
2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by $$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$ (You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
4. 1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
   2. Hence evaluate \(q\) to three decimal places. [1 mark]

The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$

1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
2. Find the mean value of \(X\). [4 marks]

A continuous random variable \(X\) has a probability density function given by $$f(x) = \frac{x^2}{312} \quad 4 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Find E\((X)\). [3 marks]
2. Find the variance of \(X\). [4 marks]
3. Find the cumulative distribution function F\((x)\), for all values of \(x\). [5 marks]
4. Hence find the median value of \(X\). [3 marks]
5. Write down the modal value of \(X\). [1 mark]

It is sometimes suggested that, for most distributions, $$2 \times (\text{median} - \text{mean}) \approx \text{mode} - \text{median}.$$

1. Show that this result is not satisfied in this case, and suggest a reason why. [2 marks]

A random variable \(X\) has a probability density function given by $$f(x) = \frac{4x^2(3-x)}{27} \quad 0 \leq x \leq 3,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Find the mode of \(X\). [3 marks]
2. Find the mean of \(X\). [3 marks]
3. Specify completely the cumulative distribution function of \(X\). [4 marks]
4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m^4 - 4m^3 + 13·5 = 0\), and hence show that \(1·84 < m < 1·85\). [4 marks]
5. What do these results suggest about the skewness of the distribution? [1 mark]

A continuous random variable \(X\) has probability density function f(x) given by $$\text{f(x)} = \frac{2x}{3} \quad 0 \leq x < 1,$$ $$\text{f(x)} = 1 - \frac{x}{3} \quad 1 \leq x \leq 3,$$ $$\text{f(x)} = 0 \quad \text{otherwise}.$$

1. Sketch the graph of f(x) for all \(x\). [3 marks]
2. Find the mean of \(X\). [5 marks]
3. Find the standard deviation of \(X\). [7 marks]
4. Show that the cumulative distribution function of \(X\) is given by $$\text{F(x)} = \frac{x^2}{3} \quad 0 \leq x < 1,$$ and find F(x) for \(1 \leq x \leq 3\). [6 marks]

Two people are playing darts. Peg hits points randomly on the circular board, whose radius is \(a\). If the distance from the centre \(O\) of the point that she hits is modelled by the variable \(R\),

1. explain why the cumulative distribution function \(F(r)\) is given by $$F(r) = 0 \quad r < 0,$$ $$F(r) = \frac{r^2}{a^2} \quad 0 \leq r \leq a,$$ $$F(r) = 1 \quad r > a.$$ [4 marks]
2. By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac{2a}{3}\). [7 marks] Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is $$F(x) = 0, \quad x < 0; \quad F(x) = \frac{x}{a}\left(2 - \frac{x}{a}\right), \quad 0 \leq x \leq a; \quad F(x) = 1, \quad x > a.$$
3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\). [5 marks]

A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$

1. Find the values of the constants \(a\) and \(k\). [4 marks]
2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]

A continuous random variable \(X\) has the probability density function $$\text{f}(x) = \frac{6x}{175} \quad 0 \leq x < 5,$$ $$\text{f}(x) = \frac{6x(10-x)}{875} \quad 5 \leq x \leq 10,$$ $$\text{f}(x) = 0 \quad \text{otherwise}.$$

1. Verify that f is a probability density function. [6 marks]
2. Write down the probability that \(X < 1\). [2 marks]
3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains. [7 marks]
4. Find the probability that \(2 < X < 7\). [2 marks]

The continuous random variable \(X\) has the following cumulative distribution function: $$F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{64}(16x - x^2), & 0 \leq x \leq 8, \\ 1, & x > 8. \end{cases}$$

1. Find \(P(X > 5)\). [2 marks]
2. Find and specify fully the probability density function \(f(x)\) of \(X\). [3 marks]
3. Sketch \(f(x)\) for all values of \(x\). [3 marks]

The continuous random variable \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}x, & 0 \leq x \leq 2, \\ \frac{1}{12}(6-x), & 2 \leq x \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [4 marks]
2. State the mode of \(X\). [1 mark]
3. Define fully the cumulative distribution function \(F(x)\) of \(X\). [9 marks]
4. Show that the median of \(X\) is 2.536, correct to 4 significant figures. [4 marks]

The continuous random variable \(X\) has the following cumulative distribution function: $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{432} x^2(x^2 - 16x + 72), & 0 \leq x \leq 6, \\ 1, & x > 6. \end{cases}$$

1. Find P(\(X < 2\)). [2 marks]
2. Find and specify fully the probability density function f(\(x\)) of \(X\). [4 marks]
3. Show that the mode of \(X\) is 2. [6 marks]
4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\). [1 mark]

A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable \(X\) and believes that \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}, & -4 \leq x \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Write down the name of this distribution. [1]
2. Define fully the cumulative distribution function F(x) of \(X\). [4]
3. Calculate the proportion of children making an error of less than 15\% according to this model. [3]
4. Give two reasons why this may not be a very suitable model. [2]

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: $$F(t) = \begin{cases} 0, & t < 0, \\ \frac{1}{135}(54t + 9t^2 - 4t^3), & 0 \leq t \leq 3, \\ 1, & t > 3. \end{cases}$$

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

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{k}{x^n} & x \geqslant 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1.

1. Find \(k\) in terms of \(n\). [3]
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\). [3]
   2. Hence determine P\((X > 7 | X > 5)\) when \(n = 4\). [4]
3. Determine the values of \(n\) for which Var\((X)\) is not defined. [5]

The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{x^3}{432}(8-x) & \text{for } 0 \leq x \leq 6, \\ 1 & \text{for } x > 6. \end{cases}$$

1. Find \(P(X > 5)\). [2]
2. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours. [2]
3. Show that the upper quartile of the distribution lies between 4·5 and 4·6. [3]
4. Find \(f(x)\), the probability density function for \(X\). [3]
5. Find the mean lifetime of the batteries in hours. [4]
6. The graph of \(f(x)\) is given below. \includegraphics{figure_1} Give a reason why the model may not be appropriate. [1]

A continuous random variable \(X\) has cumulative distribution function \(F\) given by $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{1}{4}x & \text{for } 0 \leqslant x \leqslant 2, \\ \frac{1}{480}x^4 + \frac{7}{15} & \text{for } 2 < x \leqslant b, \\ 1 & \text{for } x > b. \end{cases}$$

1. Show that \(b = 4\). [2]
2. Find P\((X \leqslant 2 \cdot 5)\). [2]
3. Write down the value of the lower quartile of \(X\). [1]
4. Find the value of the upper quartile of \(X\). [3]
5. Find, correct to three significant figures, the value of \(k\) that satisfies the equation P\((X > 3 \cdot 5) = \text{P}(X < k)\). [4]

The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function $$f(t) = \frac{1}{2500}t(100-t^2) \quad \text{for } 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise.}$$

1. Determine the mean queueing time. [3]
2. 1. Find the cumulative distribution function, \(F(t)\), of \(T\).
   2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
   3. Find the median queueing time. [10]

The length of time, in years, that a salesman keeps his company car may be modelled by the continuous random variable \(T\). The probability density function of \(T\) is given by $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$

1. Sketch the graph of \(f(t)\). [2]
2. Find the cumulative distribution function \(F(t)\) and hence find the median value of \(T\). [3]
3. Find the probability that \(T\) is greater than the modal value of \(T\). [5]
4. The probability that a randomly chosen salesman keeps his car longer than \(N\) years is \(0.05\). Find the value of \(N\) correct to \(3\) significant figures. [3]