b:head_first_statistics:using_discrete_probability_distributions

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b:head_first_statistics:using_discrete_probability_distributions [2019/10/10 18:09] hkimscil |
b:head_first_statistics:using_discrete_probability_distributions [2019/10/14 03:34] (current) hkimscil [e.g.] |
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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(Y) \\ | + | Var(X - Y) & = & Var(X) + Var(Y) \\ |

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) - b^{2}Var(Y) \\ | + | Var(aX - bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\ |

\end{eqnarray*} | \end{eqnarray*} | ||

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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? | 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? | ||

+ | <code> | ||

+ | x1 <- c(10,15,20,25) | ||

+ | x1p <- c(.2,.5,.2,.1) | ||

+ | x2 <- c(15,20,25,30) | ||

+ | x2p <- c(.15,.6,.2,.05) | ||

+ | x1n <- 25 | ||

+ | x2n <- 20 | ||

+ | x1mu <- sum(x1*x1p) | ||

+ | x2mu <- sum(x2*x2p) | ||

+ | |||

+ | x1e <- x1mu*x1num | ||

+ | x2e <- x2mu*x2num | ||

+ | |||

+ | x1e | ||

+ | x2e | ||

+ | </code> | ||

+ | |||

+ | <code>> x1e | ||

+ | [1] 400 | ||

+ | > x2e | ||

+ | [1] 415 | ||

+ | > </code> | ||

+ | 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? What’s the variance of this? | ||

+ | </WRAP> | ||

+ | | 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 | | ||

+ | |||

+ | |||

+ | <code> | ||

+ | x3 <- c(20,30,40,45) | ||

+ | x3p <- c(.3,.4,.2,.1) | ||

+ | 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 | ||

+ | |||

+ | </code> | ||

+ | |||

+ | <code> | ||

+ | > x3 <- c(20,30,40,45) | ||

+ | > x3p <- c(.3,.4,.2,.1) | ||

+ | > 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 | ||

+ | </code> | ||

b/head_first_statistics/using_discrete_probability_distributions.1570700357.txt.gz · Last modified: 2019/10/10 18:09 by hkimscil