5.03a Continuous random variables: pdf and cdf

The continuous random variable X has cumulative distribution function $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{4}x²(4 - x²), & 0 \leq x \leq 1, \\ 1, & x > 1. \end{cases}$$

1. Find P(X > 0.7). [2]
2. Find the probability density function f(x) of X. [2]
3. Calculate E(X) and show that, to 3 decimal places, Var(X) = 0.057. [6]

One measure of skewness is $$\frac{\text{Mean} - \text{Mode}}{\text{Standard deviation}}$$

1. Evaluate the skewness of the distribution of X. [4]

A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable X is the volume of lemonade dispensed into a cup.

1. Specify the probability density function of X and sketch its graph. [4]

Find the probability that the machine dispenses

1. less than 183 ml, [3]
2. exactly 183 ml. [1]
3. Calculate the inter-quartile range of X. [3]
4. Determine the value of s such that P(X ≤ s) = 1 - 2P(X ≤ s). [2]
5. Interpret in words your value of s.

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]

The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).

1. Write down fully the probability density function f(x) of \(X\). [2]
2. Sketch the probability density function f(x) of \(X\). [2]

Find

1. E(\(X^2\)), [3]
2. P(\(-0.2 < X < 0.6\)). [2]

The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the value of \(k\) is \(\frac{1}{50}\). [3]
2. Find P(\(T > 6\)). [2]
3. Calculate an exact value for E(\(T\)) and for Var(\(T\)). [5]
4. Write down the mode of the distribution of \(T\). [1]

It is suggested that the probability density function, f(\(t\)), is not a good model for \(T\).

1. Sketch the graph of a more suitable probability density function for \(T\). [1]

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]

The continuous random variable \(X\) is uniformly distributed over the interval \([-1,3]\). Find

1. E(\(X\)) [1]
2. Var(\(X\)) [2]
3. E(\(X^2\)) [2]
4. P(\(X < 1.4\)) [1]

A total of 40 observations of \(X\) are made.

1. Find the probability that at least 10 of these observations are negative. [5]

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 queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$\text{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that \(k = \frac{4}{6561}\). [3]

Using integration, find

1. the mean queuing time of a customer, [4]
2. the probability that a customer will queue for more than 5 minutes. [3]

Three independent customers shop at the post office.

1. Find the probability that at least 2 of the customers queue for more than 5 minutes. [3]

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 drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable \(X\) is the volume of lemonade dispensed into a cup.

1. Specify the probability density function of \(X\) and sketch its graph. [4]
2. Find the probability that the machine dispenses
   1. less than 183 ml,
   2. exactly 183 ml.
   [3]
3. Calculate the inter-quartile range of \(X\). [1]
4. Determine the value of \(x\) such that P(\(X \geq x\)) = 2P(\(X \leq x\)). [3]
5. Interpret in words your value of \(x\). [2]

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 \(L\) represents the error, in mm, made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval \([-4.0, 4.0]\). Find

1. P\((L < -2.6)\), [1]
2. P\((L < -3.0 \text{ or } L > 3.0)\). [2]

A random sample of 20 rods cut by the machine was checked.

1. Find the probability that more than half of them were within 3.0 mm of the target length. [4]

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]

A rectangle has a perimeter of 20 cm. The length, \(x\) cm, of one side of this rectangle is uniformly distributed between 1 cm and 7 cm. Find the probability that the length of the longer side of the rectangle is more than 6 cm long. [5]

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]

The random variable \(Y\) has probability density function f(y) given by $$\text{f}(y) = \begin{cases} ky(a - y) & 0 \leqslant y \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. 1. Explain why \(a \geqslant 3\)
   2. Show that \(k = \frac{2}{9(a-2)}\)
   [6]

Given that E(Y) = 1.75

1. show that \(a = 4\) and write down the value of \(k\). [6]

For these values of \(a\) and \(k\),

1. sketch the probability density function, [2]
2. write down the mode of \(Y\). [1]

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]

The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval [0, 0.5]

1. Find P(\(L < 0.4\)). [1]
2. Write down E(\(L\)). [1]
3. Calculate Var(\(L\)). [2]

A random sample of 30 poles cut by this machine is taken.

1. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. [3]

When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function F(\(x\)) where $$\text{F}(x) = \begin{cases} 0 & x < 0 \\ 4x - 4x^2 & 0 \leq x \leq 0.5 \\ 1 & \text{otherwise} \end{cases}$$

1. Using this model, find P(\(X > 0.4\)) [2]

A random sample of 100 poles cut by this new machine is taken.

1. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres. [3]

A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]

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]