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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
@@ Line 277: / Line 277: @@
 > </code>
-<WRAP box help>
 <WRAP col2>
 Q: So expectation is a lot like the
@@ Line 335: / Line 334: @@
 a slot machine with a larger variance, your
 overall winnings will be less reliable.
-</WRAP>
-</WRAP>
 Pool puzzle
-<WRAP box>
 \begin{eqnarray*}
 X & = & (\text{original win}) - (\text{original cost}) \\
@@ Line 350: / Line 347: @@
 & = & 5 * X + 3  \\
 \end{eqnarray*}
-</WRAP>
-<WRAP box>
 E(X) = -.77 and E(Y) = -.85. What is 5 * E(X) + 3?
@@ Line 359: / Line 353: @@
 $ 5 * E(X) + 3 = -0.85 $
 $ E(Y) = 5 * E(X) + 3 $
-</WRAP>
-<WRAP box>
 $ 5 * Var(X) = 13.4855 $
 $ 5^2 * Var(X) = 67.4275 $
 $ Var(Y) = 5^2 * Var(X) $
-</WRAP>
 \begin{eqnarray*}
@@ Line 384: / Line 376: @@
 Var(X + Y) & = & Var(X) + Var(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)  \\
 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) \\
+Var(aX - bY) & = & a^{2}Var(X) + b^{2}Var(Y) \\
 \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:  |||||
@@ Line 398: / Line 393: @@
 | 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?
+<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. ======
+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?
+| 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>
+====== 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]]