Expectation and variance from distribution

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is an unknown positive constant.

1. Find \(\mathrm { E } \left( X ^ { n } \right)\), where \(n \neq - 2\), and hence write down the value of \(\mathrm { E } ( X )\).
2. Find
   1. \(\operatorname { Var } ( X )\),
   2. \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   3. Find \(\mathrm { E } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)\) and \(\mathrm { E } \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } + X _ { 3 } ^ { 2 } \right)\), where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\). Hence construct unbiased estimators, \(T _ { 1 }\) and \(T _ { 2 }\), of \(\theta\) and \(\operatorname { Var } ( X )\) respectively, which are based on \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
   4. Find \(\operatorname { Var } \left( T _ { 2 } \right)\).

6 The table shows the probability distribution for the number of weekday (Monday to Friday) morning newspapers, \(X\), purchased by the Reed household per week.

1. Find values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. The number of weekday (Monday to Friday) evening newspapers, \(Y\), purchased by the same household per week is such that $$\mathrm { E } ( Y ) = 2.0 , \quad \operatorname { Var } ( Y ) = 1.5 \quad \text { and } \quad \operatorname { Cov } ( X , Y ) = - 0.43$$ Find values for the mean and variance of:
   1. \(S = X + Y\);
   2. \(\quad D = X - Y\).
3. The total cost per week, \(L\), of the Reed household's weekday morning and evening newspapers may be assumed to be normally distributed with a mean of \(\pounds 2.31\) and a standard deviation of \(\pounds 0.89\). The total cost per week, \(M\), of the household's weekend (Saturday and Sunday) newspapers may be assumed to be independent of \(L\) and normally distributed with a mean of \(\pounds 2.04\) and a standard deviation of \(\pounds 0.43\). Determine the probability that the total cost per week of the Reed household's newspapers is more than \(\pounds 5\).

1. The discrete random variable \(X\) has probability distribution

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution

   \(y\)	- 2	- 1	0	1	2
   \(\mathrm { P } ( Y = y )\)	\(3 a\)	\(a\)	\(b\)	\(a\)	\(c\)

   where \(a\), \(b\) and \(c\) are constants.
   For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
   The random variable \(T = 3 W - 8\)
4. Calculate \(\mathrm { P } ( W > T )\)

1. The discrete random variable \(X\) has probability distribution

where \(a\) and \(b\) are probabilities.

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 3.9\)
2. find the value of \(a\) and the value of \(b\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
3. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)\)

1. The discrete random variable \(X\) has the following probability distribution

1. Find \(\operatorname { Var } ( X )\)
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)