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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Continoups probability models
- Uniform
- Normal
- Exponential
- Gamma
- Beta
Uniform
Uniform
- Two parameters:
- \(a\), left end point
- \(b\), right end point
- All values between \(a\) and \(b\) are equally likely to occur.
Uniform
- Applications
- Generation of random variables from other distributions. (How might one generate a Bernoulli random variable from a uniform random variable?)
- Maximum uncertainty models
Normal
Exponential
Gamma
Beta
Mixture distributions
Mixture distributions (continuous and discrete)
Finite Mixture Distributions
Suppose an outcome is of the following types with corresponding probability:
| Type | Prob |
|---|---|
| 0 | .25 |
| 1 | .5 |
| 2 | .25 |
Finite Mixture Distributions
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) |
Finite Mixture Distributions
- We know the distribution of \(X|\text{Type}\).
- What is the distribution of \(X\)?
Finite Mixture Distributions
- We know the conditional distribution of \(X|\text{Type}\).
- What is the marginal distribution of \(X\)?
Finite Mixture Distributions
- 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 |
Finite Mixture Distributions
- 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 |
Finite Mixture Distributions
\[\begin{align*}P(X < x) = &P(\text{Type} = 0)P(X < x | \text{Type} = 0) \\ & + P(\text{Type} = 1)P(X < x | \text{Type} = 1)\\ & + P(\text{Type} = 2)P(X < x | \text{Type} = 2)\\ \end{align*}\\\]
Finite Mixture Distributions
px <- function(x){
.25*pnorm(x) + .5*pexp(x) + .25*punif(x,-1,1)
}
Finite Mixture Distributions
A quick note on sampling distributions
(Book section 7.2. Will come back to this.)
A quick note on sampling distributions
(Book section 7.2. Will come back to this.)
