5.03a Continuous random variables: pdf and cdf

A continuous random variable \(X\) has cumulative distribution function F given by $$\mathrm{F}(x) = \begin{cases} 0 & x < -1, \\ \frac{1}{4}(x^3 + 1) & -1 \leqslant x \leqslant 1, \\ 1 & x > 1. \end{cases}$$ Find \(\mathrm{P}\left(X \geqslant \frac{3}{4}\right)\), and state what can be deduced about the upper quartile of \(X\). [3] Obtain the cumulative distribution function of \(Y\), where \(Y = X^2\). [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 time, \(T\) seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by $$f(t) = \begin{cases} \frac{1}{100}e^{-0.01t} & t \geq 0, \\ 0 & \text{otherwise.} \end{cases}$$

1. State the expected value of \(T\). [1]
2. Find the median value of \(T\). [3]

Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes. [2]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{2} & 1 \leq x \leq 3, \\ 0 & \text{otherwise.} \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^3\). Find the distribution function of \(Y\). [5] Sketch the graph of the probability density function of \(Y\). [3] Find the probability that \(Y\) lies between its median value and its mean value. [4]

The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.

1. Write down the probability density function of \(T\). [1]
2. Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours. [3]
3. Find the median value of \(T\). [3]

The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm{F}(t) = \begin{cases} 1 - e^{-at} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200.

1. Write down the value of \(a\). [1]
2. Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]

A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99.

1. Find the smallest possible value of \(n\). [4]

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]

The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\text{P}(X < 5) = \frac{20}{39}\), find \(\text{P}(3 < X < 5)\). [2]

The probability density function, f, of a random variable \(X\) is given by $$\text{f}(x) = \begin{cases} k(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant. State the value of \(\text{E}(X)\) and show that \(\text{Var}(X) = \frac{9}{5}\). [6]

\includegraphics{figure_7} The diagram shows the graph of the probability density function, f, of a random variable \(X\) which takes values between \(-3\) and 2 only.

1. Given that the graph is symmetrical about the line \(x = -0.5\) and that P(\(X < 0\)) = \(p\), find P(\(-1 < X < 0\)) in terms of \(p\). [2]
2. It is now given that the probability density function shown in the diagram is given by $$\text{f}(x) = \begin{cases} a - b(x^2 + x) & -3 \leq x \leq 2, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are positive constants.
   1. Show that \(30a - 55b = 6\). [3]
   2. By substituting a suitable value of \(x\) into f(\(x\)), find another equation relating \(a\) and \(b\) and hence determine the values of \(a\) and \(b\). [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]

A random variable \(X\) has probability density function \(f\) defined by $$f(x) = \begin{cases} \frac{a}{x^2} - \frac{18}{x^3} & 2 \leqslant x < 3, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{27}{2}\). [3]
2. Show that \(\text{E}(X) = \frac{27}{2} \ln \frac{3}{2} - 3\). [3]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} k(1-x) & -1 \leq x \leq 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{2}\). [2]
2. Find \(\text{P}(X > \frac{1}{2})\). [1]
3. Find the mean of \(X\). [3]
4. Find \(a\) such that \(\text{P}(X < a) = \frac{1}{3}\). [3]

The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm{f}(t) = \begin{cases} k(10t - t^2) & 5 \leq t \leq 10, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{3}{250}\). [3]
2. Find \(\mathrm{E}(T)\). [3]
3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm{E}(T)\) and the median of \(T\). [3]
4. State the greatest possible length of time taken to complete the test. [1]

The average speed of a bus, \(x\) km h\(^{-1}\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$\text{f}(x) = \begin{cases} \frac{k}{x^2} & 20 \leq x \leq 28, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = 70\). [3]
2. Find E\((X)\). [3]
3. Find P\((X < \text{E}(X))\). [2]
4. Hence determine whether the mean is greater or less than the median. [2]

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

1. Show that \(k = \frac{e}{e-1}\). [3]
2. Find E(\(X\)) in terms of \(e\). [4]

The lifetimes, in hours, of light bulbs have an exponential distribution with parameter \(\frac{1}{500}\). Each bulb is tested and rejected if the lifetime is less than 500 hours.

1. Find the probability that a bulb of this type has a lifetime of more than 500 hours. [4]
2. Find the probability that the lifetime is at least three times the expected lifetime. [6]

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]

The continuous random variable \(X\) is uniformly distributed over the interval \([a, b]\) Given that \(\mathrm{P}(3 < X < 5) = \frac{1}{8}\) and \(\mathrm{E}(X) = 4\)

1. find the value of \(a\) and the value of \(b\) [3]
2. find the value of the constant, \(c\), such that \(\mathrm{E}(cX - 2) = 0\) [2]
3. find the exact value of \(\mathrm{E}(X^2)\) [3]
4. find \(\mathrm{P}(2X - b > a)\) [2]

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]

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]