Pure expectation and variance calculation

Questions that ask only to find E(aX + bY + c) and/or Var(aX + bY + c) with no further probability calculations or applications, where distributions are fully specified.

16 questions

CAIE S2 2021 June Q4
4 Wendy's journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions \(\mathrm { N } \left( 15.0,1.1 ^ { 2 } \right) , \mathrm { N } \left( 32.0,3.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 8.6,1.2 ^ { 2 } \right)\) respectively.
  1. Find the mean and variance of the total time for Wendy's journey.
    If Wendy's journey takes more than 60 minutes, she is late for work.
  2. Find the probability that, on a randomly chosen day, Wendy will be late for work.
  3. Find the probability that the mean of Wendy's journey times over 15 randomly chosen days will be less than 54.5 minutes.
CAIE S2 2022 June Q4
4 The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( 2 )\) and \(\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)\) respectively.
  1. Find the mean and standard deviation of \(X - 3 Y\).
  2. Find \(\mathrm { P } ( Y = 15 X )\).
CAIE S2 2011 June Q1
1 The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg . The total weight of a random sample of three bags is denoted by \(T \mathrm {~kg}\). Find the mean and standard deviation of \(T\).
\(2 X\) is a random variable having the distribution \(\mathrm { B } \left( 12 , \frac { 1 } { 4 } \right)\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8 .
CAIE S2 2004 June Q3
3 The independent random variables \(X\) and \(Y\) are such that \(X\) has mean 8 and variance 4.8 and \(Y\) has a Poisson distribution with mean 6. Find
  1. \(\mathrm { E } ( 2 X - 3 Y )\),
  2. \(\operatorname { Var } ( 2 X - 3 Y )\).
CAIE S2 2015 June Q7
7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find
  1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
  2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
  3. \(\mathrm { P } ( 2 X - Y = 18 )\).
CAIE S2 2015 June Q1
1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).
CAIE S2 2016 March Q1
1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).
OCR S3 2011 June Q1
1 The random variables \(X\) and \(Y\) are independent with \(X \sim \operatorname { Po } ( 5 )\) and \(Y \sim \operatorname { Po } ( 4 )\). \(S\) denotes the sum of 2 observations of \(X\) and 3 observations of \(Y\).
  1. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
  2. The random variable \(T\) is defined by \(\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y\). Show that \(\mathrm { E } ( T ) = \operatorname { Var } ( T )\).
  3. State which of \(S\) and \(T\) (if either) does not have a Poisson distribution, giving a reason for your answer.
OCR S3 2013 January Q1
1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).
  1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
  2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).
OCR S3 2012 June Q5
5 The discrete random variables \(X\) and \(Y\) are independent with \(X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)\) and \(Y \sim \operatorname { Po } ( 28 )\).
  1. Find the values of \(\mathrm { E } ( Y - X )\) and \(\operatorname { Var } ( Y - X )\).
  2. State, with justification, an approximate distribution for \(Y - X\).
  3. Hence find \(\mathrm { P } ( | Y - X | \geqslant 3 )\).
Edexcel S3 Q8
  1. The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$ where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( A )\),
  2. \(\operatorname { Var } ( A )\). The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
  3. Find \(\mathrm { P } ( B > A )\).
    Paper Reference(s) \section*{6691/01 Edexcel GCE} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-041_97_306_495_1635}
    \end{figure} \(0 - 3\) & 8
    \hline \(3 - 5\) & 12
    \hline \(5 - 6\) & 13
    \hline \(6 - 8\) & 9
    \hline \(8 - 12\) & 8
    \hline \end{tabular} \captionsetup{labelformat=empty} \caption{Table 1} \end{center} \end{table}
  4. Show that an estimate of \(\bar { X } = 5.49\) and an estimate of \(S _ { X } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
    Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
    Waiting Time\(\mathrm { x } < 3\)\(3 - 5\)\(5 - 6\)\(6 - 8\)\(\mathrm { x } > 8\)
    Expected Frequency8.5612.737.56ab
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  5. Find the value of a and the value of b .
  6. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.
    \section*{Q uestion 4 continued}
    1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
    One large and 3 small bottles of Blumen are chosen at random.
  7. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
  8. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
    \section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
  9. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
  10. Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
    7. Lambs are born in a shed on M ill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
  11. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
  12. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
  13. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
    \section*{Q uestion 7 continued}
Edexcel S3 2009 June Q8
  1. The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$ where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( A )\),
  2. \(\operatorname { Var } ( A )\). The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
  3. Find \(\mathrm { P } ( B > A )\).
Edexcel S4 2013 June Q4
  1. A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)
    1. Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias.
    The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
  2. Write down the value of the constant \(k\).
  3. Find \(\operatorname { Var } ( Y )\). The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\)
    The probability density function, \(m ( x )\), of \(M\) is given by $$m ( x ) = \left\{ \begin{array} { c l } \frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a
    0 & \text { otherwise } \end{array} \right.$$
  4. Show that \(M\) is an unbiased estimator of \(a\). Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
  5. find \(\operatorname { Var } ( M )\).
  6. State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\). A random sample of two values of \(X\) are 5 and - 1
  7. Use your answer to part (f) to estimate \(a\).
OCR MEI Further Statistics B AS 2021 November Q2
2 Natasha is investigating binomial distributions. She constructs the spreadsheet in Fig. 2 which shows the first 3 and last 4 rows of a simulation involving two independent variables, \(X\) and \(Y\), with distributions \(\mathrm { B } ( 10,0.3 )\) and \(\mathrm { B } ( 50,0.3 )\) respectively. The spreadsheet also shows the corresponding value of the random variable \(Z\), defined by \(Z = 5 X - Y\), for each pair of values of \(X\) and \(Y\). There are 100 simulated values of each of \(X , Y\) and \(Z\). The spreadsheet also shows whether each value of \(Z\) is greater than 6, and cells D103 and D104 show the number of values of \(Z\) which are greater than 6 and not greater than 6 respectively. \begin{table}[h]
1ABCDE
1XY\(\mathbf { Z } = \mathbf { 5 } \mathbf { X } - \mathbf { Y }\)\(\mathbf { Z } > \mathbf { 6 }\)
24137Y
34173N
4321-6N
5
6
98114-9N
9951213Y
100318-3N
1013150N
102
103Number of Y19
104Number of N81
105
\captionsetup{labelformat=empty} \caption{Fig. 2}
\end{table}
  1. Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( Z > 6 )\).
  2. Explain how a more reliable estimate of \(\mathrm { P } ( Z > 6 )\) could be obtained.
    1. State the greatest possible value of \(Z\).
    2. Explain why it is very unlikely that \(Z\) would have this value.
  4. Use the Central Limit Theorem to calculate an estimate of the probability that the mean of 20 independent values of \(Z\) is greater than 2 .
SPS SPS FM Statistics 2020 October Q1
  1. \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
  2. if \(X\) and \(Y\) are independent then \(\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )\).
\section*{Discrete distributions} \(X\) is a random variable taking values \(x _ { i }\) in a discrete distribution with \(\mathrm { P } \left( X = x _ { i } \right) = p _ { i }\)
Expectation: \(\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }\)
Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }\)
\(P ( X = x )\)E \(( X )\)\(\operatorname { Var } ( X )\)
Binomial \(\mathrm { B } ( n , p )\)\(\binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }\)\(n p\)\(n p ( 1 - p )\)
Uniform distribution over \(1,2 , \ldots , n , \mathrm { U } ( n )\)\(\frac { 1 } { n }\)\(\frac { n + 1 } { 2 }\)\(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\)
Geometric distribution Geo(p)\(( 1 - p ) ^ { x - 1 } p\)\(\frac { 1 } { p }\)\(\frac { 1 - p } { p ^ { 2 } }\)
Poisson \(\operatorname { Po } ( \lambda )\)\(e ^ { - \lambda } \frac { \lambda ^ { x } } { x ! }\)\(\lambda\)\(\lambda\)
\section*{Continuous distributions} \(X\) is a continuous random variable with probability density function (p.d.f.) \(\mathrm { f } ( x )\)
Expectation: \(\mu = \mathrm { E } ( X ) = \int x \mathrm { f } ( x ) \mathrm { d } x\)
Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \int ( x - \mu ) ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \int x ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x - \mu ^ { 2 }\)
Cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x ) = \int _ { - \infty } ^ { x } \mathrm { f } ( t ) \mathrm { d } t\)
p.d.f.E ( \(X\) )\(\operatorname { Var } ( X )\)
Continuous uniform distribution over [ \(a , b\) ]\(\frac { 1 } { b - a }\)\(\frac { 1 } { 2 } ( a + b )\)\(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\)
Exponential\(\lambda \mathrm { e } ^ { - \lambda x }\)\(\frac { 1 } { \lambda }\)\(\frac { 1 } { \lambda ^ { 2 } }\)
Normal \(N \left( \mu , \sigma ^ { 2 } \right)\)\(\frac { 1 } { \sigma \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } \left( \frac { x - \mu } { \sigma } \right) ^ { 2 } }\)\(\mu\)\(\sigma ^ { 2 }\)
\section*{Percentage points of the normal distribution} If \(Z\) has a normal distribution with mean 0 and variance 1 then, for each value of \(p\), the table gives the value of \(z\) such that \(P ( Z \leq z ) = p\).
\(p\)0.750.900.950.9750.990.9950.99750.9990.9995
\(z\)0.6741.2821.6451.9602.3262.5762.8073.0903.291
  1. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).
    a. Sketch the probability density function \(f ( x )\) of \(X\).
    b. State the value of \(\mathrm { P } ( X = 2 )\)
Find
c. \(\mathrm { E } ( X )\)
d. \(\operatorname { Var } ( X )\)
SPS SPS FM Statistics 2020 October Q2
2. The independent random variables \(X\) and \(Y\) are such that \(\mathrm { E } ( X ) = 20 , \mathrm { E } ( Y ) = 10\), \(\operatorname { Var } ( X ) = 5\) and \(\operatorname { Var } ( Y ) = 4\). Find:
a. \(\mathrm { E } ( 2 X - Y )\)
b. \(\operatorname { Var } ( 2 X - Y )\)