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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.]
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*}
  
Line 404: Line 404:
 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