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b:head_first_statistics:using_discrete_probability_distributions [2019/10/10 18:35] – [Fat Dan changed his prices] hkimscilb:head_first_statistics:using_discrete_probability_distributions [2019/10/14 04:00] – [e.g.] hkimscil
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(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*}
 +
 +----
 +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:  ||||| | Weekday:  |||||
Line 398: Line 401:
 | 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?
 +
 +<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
 +x3e+x4e
 +
 +
 +x3var <- sum(((x3-x3e)^2)*x3p)
 +x4var <- sum(((x4-x4e)^2)*x4p)
 +
 +x3var
 +x4var
 +x3var+x4var
 +
 + 
 +</code>
  
b/head_first_statistics/using_discrete_probability_distributions.txt · Last modified: 2023/10/04 10:29 by hkimscil

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