- \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
- 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 \(P ( Z \leq z ) = p\).
| \(p\) | 0.75 | 0.90 | 0.95 | 0.975 | 0.99 | 0.995 | 0.9975 | 0.999 | 0.9995 |
| \(z\) | 0.674 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.807 | 3.090 | 3.291 |
- 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. \(\operatorname { Var } ( X )\)