5.03e Find cdf: by integration

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

1. Find the distribution function of \(X\). [3]
2. The random variable \(Y\) is defined by \(Y = (X - 1)^3\). Find the probability density function of \(Y\). [4]
3. Find the median value of \(Y\). [3]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{3}{4x^2} + \frac{1}{4} & 1 \leqslant x \leqslant 3, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the distribution function of \(X\). [3]
2. Find the exact value of the interquartile range of \(X\). [5]

Answer only one of the following two alternatives. **EITHER** A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N, and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N. The mid-point of \(AB\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(MO = 0.2\) m. [4] The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -500y,$$ where \(y\) metres is the displacement from \(O\) in the direction \(OB\) at time \(t\) seconds, and state the period of the motion. [5] For the instant when the particle is 0.3 m from \(M\) for the first time, find

1. the speed of the particle, [2]
2. the time taken, after release, to reach this position. [3]

**OR** The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm{f}(t) = \begin{cases} \lambda\mathrm{e}^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise.} \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by F\((t) = 1 - \mathrm{e}^{-\lambda t}\). [2] The table below shows some values of F\((t)\) for the case when the mean is 20. Find the missing value. [2] It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below. Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. [3] Perform a \(\chi^2\)-test of goodness of fit, at the 5% level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions. [7]

The continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{1}{80}\left(3\sqrt{x} - \frac{8}{\sqrt{x}}\right) & 4 \leqslant x \leqslant 16, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the distribution function of \(X\). [3]

The random variable \(Y\) is defined by \(Y = \sqrt{X}\).

1. Find the probability density function of \(Y\). [3]

The random variable \(X\) has probability density function f given by $$\mathrm{f}(x) = \begin{cases} \frac{1}{30}\left(\frac{8}{x^2} + 3x^2 - 14\right) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the distribution function of \(X\). [3]

The random variable \(Y\) is defined by \(Y = X^2\).

1. Find the probability density function of \(Y\). [4]
2. Find the value of \(y\) such that \(\mathrm{P}(Y < y) = 0.8\). [3]

A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that P\((a < X < b) = p\) and P\((b < X < 3) = \frac{13}{10}p\), where \(p\) is a positive constant.

1. Show that \(p \leq \frac{5}{23}\). [1]
2. Find P\((b < X < 6 - a)\) in terms of \(p\). [2]

It is now given that the probability density function of \(X\) is \(f\), where $$f(x) = \begin{cases} \frac{1}{36}(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise}. \end{cases}$$

1. Given that \(b = 2\) and \(p = \frac{5}{81}\), find the value of \(a\). [5]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{8} & 0 \leq x < 1, \\ \frac{1}{28}(8 - x) & 1 \leq x \leq 8, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the cumulative distribution function of \(X\). [3]

1. Find the value of the constant \(a\) such that P\((X \leq a) = \frac{5}{7}\). [3]

The random variable \(Y\) is given by \(Y = \sqrt[3]{X}\).

1. Find the probability density function of \(Y\). [5]

A continuous random variable \(X\) has cumulative distribution function $$\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{4}x & 0 \leq x \leq 1 \\ \frac{1}{20}x^4 + \frac{1}{5} & 1 < x \leq d \\ 1 & x > d \end{cases}$$

1. Show that \(d = 2\) [2]
2. Find \(\mathrm{P}(X < 1.5)\) [2]
3. Write down the value of the lower quartile of \(X\) [1]
4. Find the median of \(X\) [3]
5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm{P}(X > 1.9) = \mathrm{P}(X < k)\) [4]

A continuous random variable \(X\) has probability density function $$\mathrm{f}(x) = \begin{cases} ax^2 + bx & 1 \leq x \leq 7 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(114a + 24b = 1\) [4]

Given that \(a = \frac{1}{90}\)

1. use algebraic integration to find \(\mathrm{E}(X)\) [4]
2. find the cumulative distribution function of \(X\), specifying it for all values of \(x\) [3]
3. find \(\mathrm{P}(X > \mathrm{E}(X))\) [2]
4. use your answer to part (d) to describe the skewness of the distribution. [2]

The continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{2}(-x^3 + 6x^2 - 5), & 1 \leq x \leq 4 \\ 1, & x > 4 \end{cases}$$

1. Find the probability density function f(x). [3]
2. Find the mode of X. [2]
3. Sketch f(x) for all values of x. [3]
4. Find the mean \(\mu\) of X. [3]
5. Show that F(\(\mu\)) > 0.5. [1]
6. Show that the median of X lies between the mode and the mean. [2]

Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable X represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of X and specify it fully. [2]
2. Calculate the mean time of arrival of the bus. [3]
3. Find the cumulative distribution function of X. [4]

Jean will be late for work if the bus arrives after 8.05 a.m.

1. Find the probability that Jean is late for work. [2]

A continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ kx^2 + 2kx, & 0 \leq x \leq 2, \\ 8k, & x > 2. \end{cases}$$

1. Show that \(k = \frac{1}{8}\). [1]
2. Find the median of X. [3]
3. Find the probability density function f(x). [3]
4. Sketch f(x) for all values of x. [3]
5. Write down the mode of X. [1]
6. Find E(X). [3]
7. Comment on the skewness of this distribution. [2]

The continuous random variable X has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{x}{15}, & \\ \frac{2x}{45}, & 2 < x < 7, \\ \frac{2}{9}, & 7 \leq x \leq 10, \\ 0, & otherwise. \end{cases}$$

1. Sketch f(x) for all values of x. [3]
2. Find expressions for the cumulative distribution function, F(x), for 0 ≤ x ≤ 2 and for 7 ≤ x ≤ 10. [8]
3. Find P(X ≤ 8.2). [2]
4. Find, to 3 significant figures, E(X). [4]

A continuous random variable X has probability density function f(x) where $$f(x) = \begin{cases} k(x^3 + 2x + 1), & -1 \leq x \leq 0, \\ 0, & otherwise \end{cases}$$ where k is a positive integer.

1. Show that k = 3. [4]

Find

1. E(X), [4]
2. the cumulative distribution function F(x), [4]
3. P(−0.3 < X < 0.3). [3]

The continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{3}{56}\). [3]
2. Find the cumulative distribution function F(\(x\)) for all values of \(x\). [4]
3. Evaluate E(\(X\)). [3]
4. Find the modal value of \(X\). [3]
5. Verify that the median value of \(X\) lies between 2.3 and 2.5. [3]
6. Comment on the skewness of \(X\). Justify your answer. [2]

A random variable \(X\) has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the cumulative distribution function F(x) can be written in the form \(ax^2 + bx + c\), for \(1 \leqslant x \leqslant 4\) where \(a\), \(b\) and \(c\) are constants. [3]
2. Define fully the cumulative distribution function F(x). [2]
3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. [6]

Given that the median of \(X\) is 1.88

1. describe the skewness of the distribution. Give a reason for your answer. [2]

A continuous random variable \(X\) has the probability density function f(\(x\)) shown in Figure 1. \includegraphics{figure_1} Figure 1

1. Show that f(\(x\)) = \(4 - 8x\) for \(0 \leqslant x \leqslant 0.5\) and specify f(\(x\)) for all real values of \(x\). [4]
2. Find the cumulative distribution function F(\(x\)). [4]
3. Find the median of \(X\). [3]
4. Write down the mode of \(X\). [1]
5. State, with a reason, the skewness of \(X\). [1]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{2}{15}, & 2 < x < 7, \\ \frac{4}{9} - \frac{2x}{45}, & 7 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [3]
2. 1. Find expressions for the cumulative distribution function, \(\mathrm{F}(x)\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
   2. Show that for \(2 < x < 7\), \(\mathrm{F}(x) = \frac{2x}{15} - \frac{2}{15}\).
   3. Specify \(\mathrm{F}(x)\) for \(x < 0\) and for \(x > 10\).
   [8]
3. Find \(\mathrm{P}(X \leq 8.2)\). [2]
4. Find, to 3 significant figures, \(\mathrm{E}(X)\). [4]

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

1. Show that \(k = 3\). [4]

Find

1. E(\(X\)), [4]
2. the cumulative distribution function F(\(x\)), [4]
3. P(\(-0.3 < X < 0.3\)). [3]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{3}, & 0 \leq x \leq 1, \\ \frac{8x^3}{45}, & 1 \leq x \leq 2, \\ 0, & \text{otherwise}. \end{cases}$$

1. Calculate the mean of \(X\). [5]
2. Specify fully the cumulative distribution function F\((x)\). [7]
3. Find the median of \(X\). [3]
4. Comment on the skewness of the distribution of \(X\). [2]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{1+x}{k}, & 1 \leqslant x \leqslant 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{21}{2}\). [3]
2. Specify fully the cumulative distribution function of \(X\). [5]
3. Calculate E\((X)\). [3]
4. Find the value of the median. [3]
5. Write down the mode. [1]
6. Explain why the distribution is negatively skewed. [1]

The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{4}{9}(x^2 + 2x - 3) & 1 \leqslant x \leqslant 1.5 \\ 1 & x > 1.5 \end{cases}$$

1. Find the median of \(X\), giving your answer to 3 significant figures. [3]
2. Find, in full, the probability density function of the random variable \(X\). [3]
3. Find P(\(X \geqslant 1.2\)) [2]

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.

1. Find the probability that the lantern will still be working after 12 hours. [2]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} kx^2 & 0 \leq x \leq 2 \\ k\left(1 - \frac{x}{6}\right) & 2 < x \leq 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{4}\) [4]
2. Write down the mode of \(X\). [1]
3. Specify fully the cumulative distribution function F(\(x\)). [5]
4. Find the upper quartile of \(X\). [4]

A piece of string \(AB\) has length 12 cm. A child cuts the string at a randomly chosen point \(P\), into two pieces. The random variable \(X\) represents the length, in cm, of the piece \(AP\).

1. Suggest a suitable model for the distribution of \(X\) and specify it fully [2]
2. Find the cumulative distribution function of \(X\). [4]
3. Write down P(\(X < 4\)). [1]

The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$

1. Sketch f(\(x\)) for all values of \(x\). [3]
2. Calculate E(\(X\)). [3]
3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
5. Find the interquartile range for the random variable \(X\). [4]

Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$

1. Use this formula to estimate the interquartile range in this case, and comment. [2]