5.03c Calculate mean/variance: by integration

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise. } \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

11 The time, \(T\) years, before a particular type of washing machine breaks down may be taken to have probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants. It may be assumed that, if \(n\) is a positive integer, $$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$

1. Records show that the mean of \(T\) is 1.5 . Show that \(b = \frac { 4 } { 3 }\) and find the value of \(a\).
2. Find \(\operatorname { Var } ( T )\).
3. Calculate \(\mathrm { P } ( T < 1.5 )\). State, giving a reason, whether this value indicates that the median of \(T\) is smaller than the mean of \(T\) or greater than the mean of \(T\).

The continuous random variable \(X\) has distribution function given by $$\text{F}(x) = \begin{cases} 0 & x < 0, \\ 1 - e^{-\frac{x}{4}} & x \geqslant 0. \end{cases}$$ For a random value of \(X\), find the probability that \(2\) lies between \(X\) and \(4X\). [3] Find also the expected value of the width of the interval \((X, 4X)\). [4]

The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\). [2]
2. State the value of E\((X)\). [1]
3. Find the interquartile range of \(X\). [4]

The variable \(Y\) is related to \(X\) by \(Y = 2^X\).

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

The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{2t}x^2 & 1 \leq x \leq 4, \\ 0 & \text{otherwise}. \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^2\). Show that \(Y\) has probability density function given by $$g(y) = \begin{cases} \left(\frac{1}{2t}\right)y^{\frac{1}{2}} & 1 \leq y \leq 16, \\ 0 & \text{otherwise}. \end{cases}$$ [5] Find

1. the median value of \(Y\), [2]
2. the expected value of \(Y\). [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]

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 diagram shows the graph of the probability density function of a random variable \(X\), where $$f(x) = \begin{cases} \frac{1}{6}(3x - x^2) & 0 \leq x \leq 3, \\ 0 & \text{otherwise}. \end{cases}$$ \includegraphics{figure_1}

1. State the values of E(\(X\)) and Var(\(X\)). [4]
2. State the values of P(\(0.5 < X < 1\)). [1]
3. Given that P(\(1 < X < 2\)) = \(\frac{13}{27}\), find P(\(X > 2\)). [2]

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 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]

In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].

1. Write down P(T < 0.2). [1]
2. Write down E(T). [1]
3. Use integration to find Var(T). [4]

A group of 20 children each play this game once.

1. Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]

The children are allowed to practise this game so that this continuous uniform model is no longer applicable.

1. Explain how you would expect the mean and variance of T to change. [2]

It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by $$g(t) = \begin{cases} 4t, & 0 \leq t \leq 0.5, \\ 4 - 4t, & 0.5 \leq t \leq 1, \\ 0, & otherwise. \end{cases}$$

1. Using this model show that P(T < 0.2) = 0.08. [2]

A group of 75 experienced players each played this game once.

1. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]

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 R is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that E(R) = 3 and Var(R) = \(\frac{4}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. P(R < 6.6). [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]

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]