Calculate Var(X) from probability function

5. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2
k ( x - 2 ) , & x = 3
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = 0.25\).
2. Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
3. Find \(\operatorname { Var } ( 3 X - 2 )\). Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
4. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
5. Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
6. Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).

4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } k \left( x ^ { 2 } - 9 \right) , & x = 4,5,6
0 , & \text { otherwise } \end{array}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Find \(\operatorname { Var } ( 2 X - 3 )\).

6 The discrete random variable \(Y\) has the probability function $$\mathrm { P } ( Y = y ) = \begin{cases} 2 k y & y = 1,2,3,4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. Show that \(\operatorname { Var } ( 5 Y - 2 ) = 25\)
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