The Yankees should be winning the World Series in class examples and homeworks. Realism is essential to help fight the over-optimistic ivory tower white washing of how the world really works.

The Yankees should be winning the World Series in class examples and homeworks. Realism is essential to help fight the over-optimistic ivory tower white washing of how the world really works.

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- Uniform
- Normal
- Exponential
- Gamma
- Beta

- Two parameters:
- \(a\), left end point
- \(b\), right end point

- All values between \(a\) and \(b\) are equally likely to occur.

- Applications
- Generation of random variables from other distributions. (How might one generate a Bernoulli random variable from a uniform random variable?)
- Maximum uncertainty models

Suppose an outcome is of the following types with corresponding probability:

Type | Prob |
---|---|

0 | .25 |

1 | .5 |

2 | .25 |

Now suppose that the distribution of outcome X depends on Type. For example,

Type | Distn \(X|\text{Type}\) |
---|---|

0 | N(0,1) |

1 | Exp(1) |

2 | Uniform(-1, 1) |

- We know the distribution of \(X|\text{Type}\).
- What is the distribution of \(X\)?

- We know the
**conditional**distribution of \(X|\text{Type}\). - What is the
**marginal**distribution of \(X\)?

- Recall:

Type 0 | Type 1 | Type 2 | ||
---|---|---|---|---|

\(X \leq x\) | \(P(X \leq x)\) | |||

\(X > x\) | \(P(X > x)\) | |||

\(p_0 = P(\text{Type } 0)\) | \(p_1 = P(\text{Type } 1)\) | \(p_2 = P(\text{Type } 2)\) | 1 |

- Recall:

Type 0 | Type 1 | Type 2 | ||
---|---|---|---|---|

\(X \leq x\) | \(p_0P(X\leq x|\text{Type } 0)\) | \(p_1P(X\leq x|\text{Type } 1)\) | \(p_2P(X\leq x|\text{Type } 2)\) | \(P(X \leq x)\) |

\(X > x\) | \(p_0P(X > x|\text{Type } 0)\) | \(p_1P(X > x|\text{Type } 1)\) | \(p_2P(X > x|\text{Type } 2)\) | \(P(X > x)\) |

\(p_0 = P(\text{Type } 0)\) | \(p_1 = P(\text{Type } 1)\) | \(p_2 = P(\text{Type } 2)\) | 1 |

\[\begin{align*}P(X < x) = &P(\text{Type} = 0)P(X < x | \text{Type} = 0) \\ & + P(\text{Type} = 1)P(X < x | \text{Type} = 1)\\ & + P(\text{Type} = 2)P(X < x | \text{Type} = 2)\\ \end{align*}\\\]

px <- function(x){ .25*pnorm(x) + .5*pexp(x) + .25*punif(x,-1,1) }

(Book section 7.2. Will come back to this.)

(Book section 7.2. Will come back to this.)