###Ejercicio 77
gas<-c(23,25,26,25,27,25,24,22,23,25,26,26,24,24,22,25,26,24,24,24,27,23)
fi<-hist(gas);fi

## $breaks
## [1] 22 23 24 25 26 27
## 
## $counts
## [1] 5 6 5 4 2
## 
## $density
## [1] 0.22727273 0.27272727 0.22727273 0.18181818 0.09090909
## 
## $mids
## [1] 22.5 23.5 24.5 25.5 26.5
## 
## $xname
## [1] "gas"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
fr<-hist(gas,freq = F)

##Ejercicio 78
force<-c(243,236,389,628,143,417,205,404,464,605,137,123,372,439,497,500,535,
         577,441,231,675,132,196,217,660,569,865,725,457,347)
hist(force,breaks =3)

hist(force,breaks =6)

hist(force,breaks =11)

##Ejercicio 79 
force<-c(311,138,340,199,270,255,332,279,231,
         296,198,296,198,269,257,236,313,281,
         288,225,216,250,259,323,280,205,279,
         159,276,354,278,221,192,281,204,361,
         321,282,254,273,334,172,240,327,261,
         282,208,213,299,318,356,269,355,275,
         234,267,240,331,222,370,226)
hist(force,breaks =6)

hist(force,breaks =9)

hist(force,breaks =17)

##Ejercicio 89
x<-rnorm(1000,10)
mean(x)
## [1] 9.992921
hist(x)

##Si parecen estar uniformemente distribuidos en la media deseada como se 
#observa en el grafico 
##Ejercicio 90
x<-rnorm(1000,20,2)
mean(x)
## [1] 20.04376
var(x)
## [1] 3.94551
hist(x)

##Ejercicio 91 
x=rnorm(100,10,sqrt(2))
y=rnorm(100,15,sqrt(3))
(mean(x)+mean(y))
## [1] 25.34119
mean(x+y)
## [1] 25.34119
z<-x+y
var(z)
## [1] 5.926666
var(x)+var(y)
## [1] 5.789811
hist(z)

#1000
x=rnorm(1000,10,sqrt(2))
y=rnorm(1000,15,sqrt(3))
(mean(x)+mean(y))
## [1] 25.03976
mean(x+y)
## [1] 25.03976
z<-x+y
var(z)
## [1] 4.811663
var(x)+var(y)
## [1] 5.078332
hist(z)

##5000
x=rnorm(5000,10,sqrt(2))
y=rnorm(5000,15,sqrt(3))
(mean(x)+mean(y))
## [1] 25.0176
mean(x+y)
## [1] 25.0176
z<-x+y
var(z)
## [1] 5.021921
var(x)+var(y)
## [1] 4.993722
hist(z)

### Ejercicio 92
x=rnorm(100,10,sqrt(2))
y=rnorm(100,15,sqrt(3))
z=x*y
mean(x)
## [1] 9.903456
var(x)
## [1] 1.914016
#1000
x=rnorm(1000,10,sqrt(2))
y=rnorm(1000,15,sqrt(3))
z=x*y
mean(x)*mean(y)
## [1] 150.6333
var(x)*var(y)
## [1] 6.338929
#5000
x=rnorm(5000,10,sqrt(2))
y=rnorm(5000,15,sqrt(3))
z=x*y
mean(x)*mean(y)
## [1] 150.0843
var(x)*var(y)
## [1] 6.081887

##Ejercicio 93 
x=rnorm(100,0,sqrt(4))
y=x^2
mean(y)
## [1] 4.565403
mean(x)^2
## [1] 0.00435629
sqrt(var(y))
## [1] 5.800012
var(x)
## [1] 4.607118
#1000
x=rnorm(1000,0,sqrt(4))
y=x^2
mean(y)
## [1] 4.217999
mean(x)^2
## [1] 0.01138866
sqrt(var(y))
## [1] 6.038088
var(x)
## [1] 4.210821
##5000
x=rnorm(5000,0,sqrt(4))
y=x^2
mean(y)
## [1] 4.099221
mean(x)^2
## [1] 0.0007327023
sqrt(var(y))
## [1] 5.956689
var(x)
## [1] 4.099308

###Ejercicio 95 
x<-1:12
y<-x[-5]
time<-y[-8]
temperature<-c(10,12,18,24,21,20,18,15,13,8)
plot(time, temperature)
points(approx(time, temperature), col = 2, pch = "*")

w<-approx(time, temperature)
w
## $x
##  [1]  1.000000  1.224490  1.448980  1.673469  1.897959  2.122449  2.346939
##  [8]  2.571429  2.795918  3.020408  3.244898  3.469388  3.693878  3.918367
## [15]  4.142857  4.367347  4.591837  4.816327  5.040816  5.265306  5.489796
## [22]  5.714286  5.938776  6.163265  6.387755  6.612245  6.836735  7.061224
## [29]  7.285714  7.510204  7.734694  7.959184  8.183673  8.408163  8.632653
## [36]  8.857143  9.081633  9.306122  9.530612  9.755102  9.979592 10.204082
## [43] 10.428571 10.653061 10.877551 11.102041 11.326531 11.551020 11.775510
## [50] 12.000000
## 
## $y
##  [1] 10.000000 10.448980 10.897959 11.346939 11.795918 12.734694 14.081633
##  [8] 15.428571 16.775510 18.122449 19.469388 20.816327 22.163265 23.510204
## [15] 23.785714 23.448980 23.112245 22.775510 22.438776 22.102041 21.765306
## [22] 21.428571 21.091837 20.836735 20.612245 20.387755 20.163265 19.877551
## [29] 19.428571 18.979592 18.530612 18.081633 17.724490 17.387755 17.051020
## [36] 16.714286 16.377551 16.040816 15.704082 15.367347 15.030612 14.591837
## [43] 14.142857 13.693878 13.244898 12.489796 11.367347 10.244898  9.122449
## [50]  8.000000
#5, 22.4°C
#9, 16,37°C

##Ejercicio 96
#Tuesday
x<-1:5
hour<-x[-2]
temperature<-c(15,14,15,18)
plot(hour, temperature)
points(approx(hour,temperature), col = 2, pch = "*")

w<-approx(hour, temperature)
w
## $x
##  [1] 1.000000 1.081633 1.163265 1.244898 1.326531 1.408163 1.489796
##  [8] 1.571429 1.653061 1.734694 1.816327 1.897959 1.979592 2.061224
## [15] 2.142857 2.224490 2.306122 2.387755 2.469388 2.551020 2.632653
## [22] 2.714286 2.795918 2.877551 2.959184 3.040816 3.122449 3.204082
## [29] 3.285714 3.367347 3.448980 3.530612 3.612245 3.693878 3.775510
## [36] 3.857143 3.938776 4.020408 4.102041 4.183673 4.265306 4.346939
## [43] 4.428571 4.510204 4.591837 4.673469 4.755102 4.836735 4.918367
## [50] 5.000000
## 
## $y
##  [1] 15.00000 14.95918 14.91837 14.87755 14.83673 14.79592 14.75510
##  [8] 14.71429 14.67347 14.63265 14.59184 14.55102 14.51020 14.46939
## [15] 14.42857 14.38776 14.34694 14.30612 14.26531 14.22449 14.18367
## [22] 14.14286 14.10204 14.06122 14.02041 14.04082 14.12245 14.20408
## [29] 14.28571 14.36735 14.44898 14.53061 14.61224 14.69388 14.77551
## [36] 14.85714 14.93878 15.06122 15.30612 15.55102 15.79592 16.04082
## [43] 16.28571 16.53061 16.77551 17.02041 17.26531 17.51020 17.75510
## [50] 18.00000
#Hour 2, 14.469°C
#Thrusday
x<-1:5
hour<-x[-3]
temperature<-c(16,11,15,20)
plot(hour, temperature)
points(approx(hour,temperature), col = 2, pch = "*")

w<-approx(hour, temperature)
w
## $x
##  [1] 1.000000 1.081633 1.163265 1.244898 1.326531 1.408163 1.489796
##  [8] 1.571429 1.653061 1.734694 1.816327 1.897959 1.979592 2.061224
## [15] 2.142857 2.224490 2.306122 2.387755 2.469388 2.551020 2.632653
## [22] 2.714286 2.795918 2.877551 2.959184 3.040816 3.122449 3.204082
## [29] 3.285714 3.367347 3.448980 3.530612 3.612245 3.693878 3.775510
## [36] 3.857143 3.938776 4.020408 4.102041 4.183673 4.265306 4.346939
## [43] 4.428571 4.510204 4.591837 4.673469 4.755102 4.836735 4.918367
## [50] 5.000000
## 
## $y
##  [1] 16.00000 15.59184 15.18367 14.77551 14.36735 13.95918 13.55102
##  [8] 13.14286 12.73469 12.32653 11.91837 11.51020 11.10204 11.12245
## [15] 11.28571 11.44898 11.61224 11.77551 11.93878 12.10204 12.26531
## [22] 12.42857 12.59184 12.75510 12.91837 13.08163 13.24490 13.40816
## [29] 13.57143 13.73469 13.89796 14.06122 14.22449 14.38776 14.55102
## [36] 14.71429 14.87755 15.10204 15.51020 15.91837 16.32653 16.73469
## [43] 17.14286 17.55102 17.95918 18.36735 18.77551 19.18367 19.59184
## [50] 20.00000
#Hour 3, 13.08

###Ejercicio 97 
x<-seq(0,2.25,0.25)
y<-c(1.2,1.18,1.1,1,0.92,0.8,0.7,0.55,0.35,0)
plot(x,y, type="p")
points(spline(x,y),type="l",col=2)

##Ejercicio 98 
t<-0:10
Tem<-c(72.5,78.1,86.4,92.3,110.6,111.5,109.5,110.2,110.5,109.9,110.2)
plot(t,Tem,type="p")
points(spline(t,Tem),type="l",col=2)
lm(Tem~t)
## 
## Call:
## lm(formula = Tem ~ t)
## 
## Coefficients:
## (Intercept)            t  
##      80.941        3.843
curve(80.941+3.843*x,add=T)

##Con las dos se calculan los valores esperados 

##Ejercicio 99 
5*10+((7*10^3)/3)-23.66667
## [1] 2359.667
#Posicion a 10 seg 2359.667
##Por integración y despeje 

library(Ryacas)

x <- Sym("x")
Integrate(5*x^2-6*x+8, x)
## expression(5 * x^3/3 - 3 * x^2 + 8 * x)