b:head_first_statistics:using_discrete_probability_distributions
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b:head_first_statistics:using_discrete_probability_distributions [2019/10/10 18:29] – [Fat Dan changed his prices] hkimscil | b:head_first_statistics:using_discrete_probability_distributions [2019/10/14 04:00] – [e.g.] hkimscil | ||
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Line 384: | Line 384: | ||
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) & = & a2Var(X) + b2Var(Y) \\ | + | Var(aX + bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\ |
+ | E(aX - bY) & = & aE(X) - bE(Y) \\ | ||
+ | Var(aX - bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\ | ||
\end{eqnarray*} | \end{eqnarray*} | ||
- | | x | -5 | + | ---- |
- | | P(X = x) | 0.99 | 0.01 | | + | 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 | | ||
+ | | P(X = x) | 0.2 | 0.5 | 0.2 | 0.1 | | ||
+ | |||
+ | | Weekend: | ||
+ | | y | 15 | 20 | 25 | 30 | | ||
+ | | 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,.5,.2,.1) | ||
+ | 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. ====== | ||
+ | <WRAP box> | ||
+ | |||
+ | 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 | ||
+ | |||
+ | | Restaurant B: | ||
+ | | y | 10 | 15 | 18 | | ||
+ | | P(Y = y) | 0.2 | 0.6 | 0.2 | | ||
+ | |||
+ | |||
+ | < | ||
+ | x3 <- c(20, | ||
+ | x3p <- c(.3, | ||
+ | x4 <- c(10, | ||
+ | x4p <- c(.2, | ||
+ | |||
+ | x3e <- sum(x3*x3p) | ||
+ | x4e <- sum(x4*x4p) | ||
+ | |||
+ | x3e | ||
+ | x4e | ||
+ | x3e+x4e | ||
- | | y | -2 | 23 | 48 | 73 | 98 | ||
- | | P(Y = y) | 0.977 | 0.008 | 0.008 | 0.006 | 0.001 | | ||
+ | x3var <- sum(((x3-x3e)^2)*x3p) | ||
+ | x4var <- sum(((x4-x4e)^2)*x4p) | ||
+ | x3var | ||
+ | x4var | ||
+ | x3var+x4var | ||
+ | |||
+ | </ | ||
b/head_first_statistics/using_discrete_probability_distributions.txt · Last modified: 2023/10/04 10:29 by hkimscil