Note: In following table, $A_i$s are positive probability events that form a partition of sample space unless otherwise stated.

Description | Discrete | Continuous |
---|---|---|

Basic Expectation Related | ||

E[X] | $\sum_x xp_X(x)$ | $\int xf_X(x)dx$ |

E[g(X)] | $\sum_x g(x)p_X(x)$ | $\int g(x)f_X(x)dx$ |

if Y = aX + b | E[Y] = aE[X] + b | E[Y] = aE[X] + b |

if Y = aX + b | var(Y) = $a^2$var(X) | var(Y) = $a^2$var(X) |

Joint PMF\PDFs | ||

Marginal PMF\PDFs | $p_X(x)$ = $\sum_y p_{X,Y}(x,y)$ | $f_X(x)$ = $\int f_{X,Y}(x,y)dy$ |

E[g(X,Y)] | $\sum_x \sum_y g(x,y)p_{X,Y}(x,y)$ | $\int \int g(x,y)f_{X,Y}(x,y)dxdy$ |

if g(X,Y) = aX + bY + c | E[g(X,Y)] = aE[X] + bE[Y] + c | E[g(X,Y)] = aE[X] + bE[Y] + c |

Conditional PMF\PDF | $p_{X,Y}(x,y)$ = $p_Y(y)p_{X|Y}(x|y)$ | $f_{X,Y}(x,y)$ = $f_Y(y)f_{X|Y}(x|y)$ |

Conditional Expectation | ||

E[X|A] | $\sum_x xp_{X|A}(x)$ | $\int xf_{X|A}(x)dx$ |

E[g(X)|A] | $\sum_x g(x)p_{X|A}(x)$ | $\int g(x)f_{X|A}(x)dx$ |

E[X|Y=y] | $\sum_x xp_{X|Y}(x|y)$ | $\int xf_{X|Y}(x|y)dx$ |

E[g(X)|Y=y] | $\sum_x g(x)p_{X|Y}(x|y)$ | $\int g(x)f_{X|Y}(x|y)dx$ |

E[g(X,Y)|Y=y] | $\sum_x g(x,y)p_{X|Y}(x|y)$ | $\int g(x,y)f_{X|Y}(x|y)dx$ |

Total Probability Related | ||

$p_X(x)$ = $\sum_{i = 1}^n P(A_i)p_{X|A_i}(x)$ | $f_X(x)$ = $\sum_{i = 1}^n P(A_i)f_{X|A_i}(x)$ | |

P(A) | $\sum_x P(A|X=x)p_X(x)$ | $\int P(A|X=x)f_X(x)dx$ |

Total Expectation Related | ||

E[X] | $\sum_y p_Y(y)E[X|Y=y]$ | $\int f_Y(y)E[X|Y=y]$ |

E[g(X)] | $\sum_y p_Y(y)E[g(X)|Y=y]$ | $\int f_Y(y)E[g(X)|Y=y]$ |

E[g(X,Y)] | $\sum_y p_Y(y)E[g(X,Y)|Y=y]$ | $\int f_Y(y)E[g(X,Y)|Y=y]$ |

E[X] | $\sum_{i=1}^n P(A_i)E[X|A_i]$ | $\sum_{i=1}^n P(A_i)E[X|A_i]$ |

E[g(X)] | $\sum_{i=1}^n P(A_i)E[g(X)|A_i]$ | $\sum_{i=1}^n P(A_i)E[g(X)|A_i]$ |

$A_i$s, partition on B | E[X|B] = $\sum_{i=1}^n P(A_i|B)E[X|A_i\cap B]$ | |

X and Y are independent | ||

$p_{X,Y}$ = $p_X(x)p_Y(y)$ | $f_{X,Y}$ = $f_X(x)f_Y(y)$ | |

E[XY] | E[X]E[Y] | E[X]E[Y] |

E[g(X)h(Y)] | E[g(X)]E[h(Y)] | E[g(X)]E[h(Y)] |

var(X + Y) | var(X) + var(Y) | var(X) + var(Y) |

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