Moment generating function problems

5 The discrete random variable \(X\) has moment generating function \(\frac { 1 } { 4 } \mathrm { e } ^ { 2 t } + a \mathrm { e } ^ { 3 t } + b \mathrm { e } ^ { 4 t }\), where \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( X ) = 3 \frac { 3 } { 8 }\).

1. Show that \(a = \frac { 1 } { 8 }\), and find the value of \(b\).
2. Find \(\operatorname { Var } ( X )\).
3. State the possible values of \(X\).

4 The discrete random variable \(X\) has moment generating function \(\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }\).

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X = 2 )\).
3. Show that \(X\) can be expressed as a sum of 3 independent observations of a random variable \(Y\). Obtain the probability distribution of \(Y\), and the variance of \(Y\).

5 The discrete random variable \(X\) is such that \(\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots\).

1. Show that the moment generating function of \(X , \mathrm { M } _ { X } ( t )\), can be written as \(\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }\).
2. Find the range of values of \(t\) for which the formula for \(\mathrm { M } _ { X } ( t )\) in part (i) is valid.
3. Use \(\mathrm { M } _ { X } ( t )\) to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 An industrial process produces components. Some of the components contain faults. The number of faults in a component is modelled by the random variable \(X\) with probability function $$\mathrm { P } ( X = x ) = \theta ( 1 - \theta ) ^ { x } \quad \text { for } x = 0,1,2 , \ldots$$ where \(\theta\) is a parameter with \(0 < \theta < 1\). The numbers of faults in different components are independent.
A random sample of \(n\) components is inspected. \(n _ { 0 }\) are found to have no faults, \(n _ { 1 }\) to have one fault and the remainder \(\left( n - n _ { 0 } - n _ { 1 } \right)\) to have two or more faults.

1. Find \(\mathrm { P } ( X \geqslant 2 )\) and hence show that the likelihood is $$\mathrm { L } ( \theta ) = \theta ^ { n _ { 0 } + n _ { 1 } } ( 1 - \theta ) ^ { 2 n - 2 n _ { 0 } - n _ { 1 } }$$
2. Find the maximum likelihood estimator \(\hat { \theta }\) of \(\theta\). You are not required to verify that any turning point you locate is a maximum.
3. Show that \(\mathrm { E } ( X ) = \frac { 1 - \theta } { \theta }\). Deduce that another plausible estimator of \(\theta\) is \(\tilde { \theta } = \frac { 1 } { 1 + \bar { X } }\) where \(\bar { X }\) is the sample mean. What additional information is needed in order to calculate the value of this estimator?
4. You are given that, in large samples, \(\tilde { \theta }\) may be taken as Normally distributed with mean \(\theta\) and variance \(\theta ^ { 2 } ( 1 - \theta ) / n\). Use this to obtain a \(95 \%\) confidence interval for \(\theta\) for the case when 100 components are inspected and it is found that 92 have no faults, 6 have one fault and the remaining 2 have exactly four faults each.

1 Traffic engineers are studying the flow of vehicles along a road. At an initial stage of the investigation, they assume that the average flow remains the same throughout the working day. An automatic counter records the number of vehicles passing a certain point per minute during the working day. A random sample of these records is selected; the sample values are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\).

1. The engineers model the underlying random variable \(X\) by a Poisson distribution with unknown parameter \(\theta\). Obtain the likelihood of \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\) and hence find the maximum likelihood estimate of \(\theta\).
2. Write down the maximum likelihood estimate of the probability that no vehicles pass during a minute.
3. The engineers note that, in a sample of size 1000 with sample mean \(\bar { x } = 5\), there are no observations of zero. Suggest why this might cast some doubt on the investigation.
4. On checking the automatic counter, the engineers find that, due to a fault, no record at all is made if no vehicle passes in a minute. They therefore model \(X\) as a Poisson random variable, again with an unknown parameter \(\theta\), except that the value \(x = 0\) cannot occur. Show that, under this model, $$\mathrm { P } ( X = x ) = \frac { \theta ^ { x } } { \left( \mathrm { e } ^ { \theta } - 1 \right) x ! } , \quad x = 1,2 , \ldots$$ and hence show that the maximum likelihood estimate of \(\theta\) satisfies the equation $$\frac { \theta \mathrm { e } ^ { \theta } } { \mathrm { e } ^ { \theta } - 1 } = \bar { x }$$

4 A biased coin has a probability \(p\) of landing on heads, where \(0 < p < 1\) Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads. Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of \(p\).
Simon proposes the estimator \(S = \frac { X + Y } { m + n }\) and Taruni proposes \(T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]\)

1. Show that both \(S\) and \(T\) are unbiased estimators of \(p\).
2. Prove that, for all values of \(m\) and \(n , S\) is the better estimator.

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ 0,4 \beta ]\), where \(\beta\) is an unknown constant.

Three independent observations, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are taken of \(X\) and the following estimators for \(\beta\) are proposed $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } } { 2 }
& B = \frac { X _ { 1 } + 2 X _ { 2 } + 3 X _ { 3 } } { 8 }
& C = \frac { X _ { 1 } + 2 X _ { 2 } - X _ { 3 } } { 8 } \end{aligned}$$

1. Calculate the bias of \(A\), the bias of \(B\) and the bias of \(C\)
2. By calculating the variances, explain which of \(B\) or \(C\) is the better estimator for \(\beta\)
3. Find an unbiased estimator for \(\beta\)

1. Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\)
   \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\)
   Korhan's estimator is given by

$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$

1. Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
2. 1. Find \(\operatorname { Var } ( K )\)
   2. Find \(\operatorname { Var } ( L )\) The winner of the challenge is the person who finds the better estimator.
3. Determine the winner of the challenge for large values of \(n\). Give reasons for your answer.

1. A bag contains a large number of marbles of which an unknown proportion, \(p\), is yellow.

Three random samples of size \(n\) are taken, and the number of yellow marbles in each sample, \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\), is recorded. Two estimators \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are proposed to estimate the value of \(p\) $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n }
& \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$

1. Show that \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are both unbiased estimators of \(p\)
2. Find the variance of \(\hat { p } _ { 1 }\) The variance of \(\hat { \mathrm { p } } _ { 2 }\) is \(\frac { 7 p ( 1 - p ) } { 18 n }\)
3. State, giving a reason, which is the better estimator. The estimator \(\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }\) where \(a\) and \(b\) are positive integers.
4. Find the pair of values of \(a\) and \(b\) such that \(\hat { \mathrm { p } } _ { 3 }\) is a better unbiased estimator of \(p\) than both \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\)
   You must show all stages of your working.