b:head_first_statistics:using_discrete_probability_distributions
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b:head_first_statistics:using_discrete_probability_distributions [2019/10/10 18:35] – [Fat Dan changed his prices] hkimscil | b:head_first_statistics:using_discrete_probability_distributions [2019/10/15 11:10] – hkimscil | ||
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> </ | > </ | ||
- | <WRAP box help> | ||
<WRAP col2> | <WRAP col2> | ||
Q: So expectation is a lot like the | Q: So expectation is a lot like the | ||
Line 335: | Line 334: | ||
a slot machine with a larger variance, your | a slot machine with a larger variance, your | ||
overall winnings will be less reliable. | overall winnings will be less reliable. | ||
- | </ | ||
- | </ | ||
Pool puzzle | Pool puzzle | ||
- | <WRAP box> | + | |
\begin{eqnarray*} | \begin{eqnarray*} | ||
X & = & (\text{original win}) - (\text{original cost}) \\ | X & = & (\text{original win}) - (\text{original cost}) \\ | ||
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& = & 5 * X + 3 \\ | & = & 5 * X + 3 \\ | ||
\end{eqnarray*} | \end{eqnarray*} | ||
- | </ | ||
- | |||
- | <WRAP box> | ||
E(X) = -.77 and E(Y) = -.85. What is 5 * E(X) + 3? | E(X) = -.77 and E(Y) = -.85. What is 5 * E(X) + 3? | ||
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$ 5 * E(X) + 3 = -0.85 $ | $ 5 * E(X) + 3 = -0.85 $ | ||
$ E(Y) = 5 * E(X) + 3 $ | $ E(Y) = 5 * E(X) + 3 $ | ||
- | </ | ||
- | <WRAP box> | + | |
$ 5 * Var(X) = 13.4855 $ | $ 5 * Var(X) = 13.4855 $ | ||
$ 5^2 * Var(X) = 67.4275 $ | $ 5^2 * Var(X) = 67.4275 $ | ||
$ Var(Y) = 5^2 * Var(X) $ | $ Var(Y) = 5^2 * Var(X) $ | ||
- | </ | ||
\begin{eqnarray*} | \begin{eqnarray*} | ||
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Var(X + Y) & = & Var(X) + Var(Y) \\ | Var(X + Y) & = & Var(X) + Var(Y) \\ | ||
E(X - Y) & = & E(X) - E(Y) \\ | E(X - Y) & = & E(X) - E(Y) \\ | ||
- | Var(X - Y) & = & Var(X) | + | Var(X - Y) & = & Var(X) |
E(aX + bY) & = & aE(X) + bE(Y) \\ | E(aX + bY) & = & aE(X) + bE(Y) \\ | ||
Var(aX + bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\ | Var(aX + bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\ | ||
E(aX - bY) & = & aE(X) - bE(Y) \\ | E(aX - bY) & = & aE(X) - bE(Y) \\ | ||
- | Var(aX - bY) & = & a^{2}Var(X) | + | Var(aX - bY) & = & a^{2}Var(X) |
\end{eqnarray*} | \end{eqnarray*} | ||
- | | Weekday: | + | ---- |
+ | A restaurant offers two menus, one for weekdays and the other for weekends. Each menu offers four set prices, and the probability distributions for the amount someone pays is as follows: | ||
+ | |||
+ | | Weekday: | ||
| x | 10 | 15 | 20 | 25 | | | x | 10 | 15 | 20 | 25 | | ||
| P(X = x) | 0.2 | 0.5 | 0.2 | 0.1 | | | P(X = x) | 0.2 | 0.5 | 0.2 | 0.1 | | ||
- | | Weekend: | + | | Weekend: |
| y | 15 | 20 | 25 | 30 | | | y | 15 | 20 | 25 | 30 | | ||
| P(Y = y) | 0.15 | 0.6 | 0.2 | 0.05 | | | P(Y = y) | 0.15 | 0.6 | 0.2 | 0.05 | | ||
+ | |||
+ | Who would you expect to pay the restaurant most: a group of 20 eating at the weekend, or a group of 25 eating on a weekday? | ||
+ | |||
+ | < | ||
+ | x1 <- c(10, | ||
+ | x1p <- c(.2, | ||
+ | x2 <- c(15, | ||
+ | x2p <- c(.15, | ||
+ | x1n <- 25 | ||
+ | x2n <- 20 | ||
+ | |||
+ | x1mu <- sum(x1*x1p) | ||
+ | x2mu <- sum(x2*x2p) | ||
+ | |||
+ | x1e <- x1mu*x1num | ||
+ | x2e <- x2mu*x2num | ||
+ | |||
+ | x1e | ||
+ | x2e | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | [1] 400 | ||
+ | > x2e | ||
+ | [1] 415 | ||
+ | > </ | ||
+ | x2e will spend more. | ||
+ | |||
+ | ====== e.g. ====== | ||
+ | |||
+ | Sam likes to eat out at two restaurants. Restaurant A is generally more expensive than | ||
+ | restaurant B, but the food quality is generally much better. | ||
+ | Below you’ll find two probability distributions detailing how much Sam tends to spend at each | ||
+ | restaurant. As a general rule, what would you say is the difference in price between the two | ||
+ | restaurants? | ||
+ | |||
+ | | Restaurant A: | ||
+ | | x | 20 | 30 | 40 | 45 | | ||
+ | | P(X = x) | 0.3 | 0.4 | 0.2 | 0.1 | | ||
+ | |||
+ | | Restaurant B: |||| | ||
+ | | y | 10 | 15 | 18 | | ||
+ | | P(Y = y) | 0.2 | 0.6 | 0.2 | | ||
+ | |||
+ | |||
+ | < | ||
+ | x3 <- c(20, | ||
+ | x3p <- c(.3, | ||
+ | x4 <- c(10,15,18) | ||
+ | x4p <- c(.2,.6,.2) | ||
+ | |||
+ | x3e <- sum(x3*x3p) | ||
+ | x4e <- sum(x4*x4p) | ||
+ | |||
+ | x3e | ||
+ | x4e | ||
+ | ## difference in price between the two | ||
+ | x3e-x4e | ||
+ | |||
+ | |||
+ | x3var <- sum(((x3-x3e)^2)*x3p) | ||
+ | x4var <- sum(((x4-x4e)^2)*x4p) | ||
+ | |||
+ | x3var | ||
+ | x4var | ||
+ | ## difference in variance between the two | ||
+ | ## == variance range | ||
+ | x3var+x4var | ||
+ | |||
+ | </ | ||
+ | |||
+ | < | ||
+ | > x3 <- c(20, | ||
+ | > x3p <- c(.3, | ||
+ | > x4 <- c(10,15,18) | ||
+ | > x4p <- c(.2,.6,.2) | ||
+ | > | ||
+ | > x3e <- sum(x3*x3p) | ||
+ | > x4e <- sum(x4*x4p) | ||
+ | > | ||
+ | > x3e | ||
+ | [1] 30.5 | ||
+ | > x4e | ||
+ | [1] 14.6 | ||
+ | > ## difference in price between the two | ||
+ | > x3e-x4e | ||
+ | [1] 15.9 | ||
+ | > | ||
+ | > | ||
+ | > x3var <- sum(((x3-x3e)^2)*x3p) | ||
+ | > x4var <- sum(((x4-x4e)^2)*x4p) | ||
+ | > | ||
+ | > x3var | ||
+ | [1] 72.25 | ||
+ | > x4var | ||
+ | [1] 6.64 | ||
+ | > ## difference in variance between the two | ||
+ | > ## == variance range | ||
+ | > x3var+x4var | ||
+ | [1] 78.89 | ||
+ | </ | ||
+ | |||
+ | ====== e.g. ====== | ||
+ | | E(aX + b) | $aE(X) + b$ | | ||
+ | | Var(aX + b) | $a^{2}Var(X)$ | ||
+ | | E(X) | $\sum{x} \cdot P(X=x) $ | | ||
+ | | E(f(X)) | $\sum{f(X)} \cdot P(X=x)$ | ||
+ | | Var(aX - bY) | $a^{2}Var(X) + b^{2}Var(Y)$ see 1 | | ||
+ | | Var(X) | $E(X-\mu)^{2} = E(X^{2})-\mu^{2}$ | ||
+ | | E(aX - bY) | $aE(X)-bE(Y)$ | ||
+ | | E(X1 + X2 + X3) | $3E(X)$ | ||
+ | | Var(X1 + X2 + X3) | $3Var(X)$ | ||
+ | | E(X2) | $\sum{X^{2}}\cdot P(X=x)$ | ||
+ | | Var(aX - b) | $a^{2}Var(X)$ | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | Var(aX - bY) & = & Var(aX + -bY) \\ | ||
+ | & = & Var(aX) + Var(-bY) \\ | ||
+ | & = & a^{2}Var(X) + b^{2}Var(Y) | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | see also [[:why n-1]] | ||
b/head_first_statistics/using_discrete_probability_distributions.txt · Last modified: 2023/10/04 10:29 by hkimscil