Probability and Distribution Exercise
Olivia Visaggio
2/11/2019
A. Permutation and Combination
- Harmony was born on 03/31/1983. How many eight-digit codes could she make using the digits in her birthday?
0 (1), 3 (3), 1 (2), 9 (1), 8(1); (n!)/(k1!xk2!xk3!…xkn!)
8!/(1!)(3!)(2!)(1!)(1!) –> 8!/(3!)(2!)
factorial(8)/(factorial(2)*factorial(3))## [1] 3360- A mother duck lines her 9 ducks up behind her. In how many ways can the ducklings line up?
factorial(9)## [1] 362880(choose(16,2)/choose(20,2))## [1] 0.6315789library(MASS)
as.fractions(0.6315789)## [1] 12/19B. Binomial Distribution Commands
x ~ Bin(n,p)
- n = 10, p = 0.5; probability of getting at least 7 heads P (x=7) = (10C7) (1/2)^10
P(x>=7)= P(x=7) + P(x=8) + P (x=9) + P(x=10)
dbinom(x, n, p)
sum(dbinom(7:10, 10, 0.5))## [1] 0.171875- n = 5, p = 1.5, A winning at least 3/5 (3, 4, 5)
sum(dbinom(3:5, 5, 3/5))## [1] 0.68256- p = 0.75, claim: at least 5 out of 6 (x=5,6); claim rejected (x=1,2,3,4)
sum(dbinom(5:6, 6, 0.75))## [1] 0.53393551-(sum(dbinom(5:6, 6, 0.75)))## [1] 0.4660645C. Normal Distribution Commands
- pnorm(x, mean, sd)
pnorm(20, 25, 5.25)## [1] 0.1704519- create an array from -100 to 100 with a step-size 0.1 (name of vector is “y”)
y <- c(-100:100, step.size = 0.1)
y##
## -100.0 -99.0 -98.0 -97.0 -96.0 -95.0 -94.0
##
## -93.0 -92.0 -91.0 -90.0 -89.0 -88.0 -87.0
##
## -86.0 -85.0 -84.0 -83.0 -82.0 -81.0 -80.0
##
## -79.0 -78.0 -77.0 -76.0 -75.0 -74.0 -73.0
##
## -72.0 -71.0 -70.0 -69.0 -68.0 -67.0 -66.0
##
## -65.0 -64.0 -63.0 -62.0 -61.0 -60.0 -59.0
##
## -58.0 -57.0 -56.0 -55.0 -54.0 -53.0 -52.0
##
## -51.0 -50.0 -49.0 -48.0 -47.0 -46.0 -45.0
##
## -44.0 -43.0 -42.0 -41.0 -40.0 -39.0 -38.0
##
## -37.0 -36.0 -35.0 -34.0 -33.0 -32.0 -31.0
##
## -30.0 -29.0 -28.0 -27.0 -26.0 -25.0 -24.0
##
## -23.0 -22.0 -21.0 -20.0 -19.0 -18.0 -17.0
##
## -16.0 -15.0 -14.0 -13.0 -12.0 -11.0 -10.0
##
## -9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
##
## -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
##
## 5.0 6.0 7.0 8.0 9.0 10.0 11.0
##
## 12.0 13.0 14.0 15.0 16.0 17.0 18.0
##
## 19.0 20.0 21.0 22.0 23.0 24.0 25.0
##
## 26.0 27.0 28.0 29.0 30.0 31.0 32.0
##
## 33.0 34.0 35.0 36.0 37.0 38.0 39.0
##
## 40.0 41.0 42.0 43.0 44.0 45.0 46.0
##
## 47.0 48.0 49.0 50.0 51.0 52.0 53.0
##
## 54.0 55.0 56.0 57.0 58.0 59.0 60.0
##
## 61.0 62.0 63.0 64.0 65.0 66.0 67.0
##
## 68.0 69.0 70.0 71.0 72.0 73.0 74.0
##
## 75.0 76.0 77.0 78.0 79.0 80.0 81.0
##
## 82.0 83.0 84.0 85.0 86.0 87.0 88.0
##
## 89.0 90.0 91.0 92.0 93.0 94.0 95.0
## step.size
## 96.0 97.0 98.0 99.0 100.0 0.1- assign z <-dnorm (y, mean = 25, sd = 5.25) creating the density function
z <- dnorm (y, mean = 25, sd = 5.25)
z##
## 6.045160e-125 5.535017e-123 4.887349e-121 4.161703e-119 3.417528e-117
##
## 2.706427e-115 2.066921e-113 1.522281e-111 1.081207e-109 7.405702e-108
##
## 4.891780e-106 3.116096e-104 1.914248e-102 1.134041e-100 6.478918e-99
##
## 3.569600e-97 1.896618e-95 9.718151e-94 4.802093e-92 2.288342e-90
##
## 1.051609e-88 4.660486e-87 1.991826e-85 8.209461e-84 3.263032e-82
##
## 1.250752e-80 4.623432e-79 1.648166e-77 5.666056e-76 1.878468e-74
##
## 6.005792e-73 1.851740e-71 5.505958e-70 1.578807e-68 4.365848e-67
##
## 1.164264e-65 2.994177e-64 7.425861e-63 1.776067e-61 4.096523e-60
##
## 9.112018e-59 1.954596e-57 4.043364e-56 8.066254e-55 1.551830e-53
##
## 2.879121e-52 5.151323e-51 8.888347e-50 1.478994e-48 2.373314e-47
##
## 3.672716e-46 5.481036e-45 7.888261e-44 1.094821e-42 1.465374e-41
##
## 1.891458e-40 2.354445e-39 2.826335e-38 3.271914e-37 3.652780e-36
##
## 3.932678e-35 4.083162e-34 4.088350e-33 3.947688e-32 3.676045e-31
##
## 3.301126e-30 2.858820e-29 2.387562e-28 1.922940e-27 1.493552e-26
##
## 1.118711e-25 8.080886e-25 5.629156e-24 3.781559e-23 2.449863e-22
##
## 1.530579e-21 9.221750e-21 5.358140e-20 3.002327e-19 1.622353e-18
##
## 8.454266e-18 4.248638e-17 2.059050e-16 9.623373e-16 4.337416e-15
##
## 1.885289e-14 7.902569e-14 3.194493e-13 1.245314e-12 4.681649e-12
##
## 1.697314e-11 5.934293e-11 2.000870e-10 6.505973e-10 2.040087e-09
##
## 6.169197e-09 1.799085e-08 5.059623e-08 1.372233e-07 3.589061e-07
##
## 9.052677e-07 2.201996e-06 5.165346e-06 1.168491e-05 2.549147e-05
##
## 5.362997e-05 1.088087e-04 2.128937e-04 4.017032e-04 7.309558e-04
##
## 1.282686e-03 2.170664e-03 3.542487e-03 5.575287e-03 8.461930e-03
##
## 1.238554e-02 1.748251e-02 2.379775e-02 3.124001e-02 3.954849e-02
##
## 4.828273e-02 5.684562e-02 6.454246e-02 7.067036e-02 7.462295e-02
##
## 7.598901e-02 7.462295e-02 7.067036e-02 6.454246e-02 5.684562e-02
##
## 4.828273e-02 3.954849e-02 3.124001e-02 2.379775e-02 1.748251e-02
##
## 1.238554e-02 8.461930e-03 5.575287e-03 3.542487e-03 2.170664e-03
##
## 1.282686e-03 7.309558e-04 4.017032e-04 2.128937e-04 1.088087e-04
##
## 5.362997e-05 2.549147e-05 1.168491e-05 5.165346e-06 2.201996e-06
##
## 9.052677e-07 3.589061e-07 1.372233e-07 5.059623e-08 1.799085e-08
##
## 6.169197e-09 2.040087e-09 6.505973e-10 2.000870e-10 5.934293e-11
##
## 1.697314e-11 4.681649e-12 1.245314e-12 3.194493e-13 7.902569e-14
##
## 1.885289e-14 4.337416e-15 9.623373e-16 2.059050e-16 4.248638e-17
##
## 8.454266e-18 1.622353e-18 3.002327e-19 5.358140e-20 9.221750e-21
##
## 1.530579e-21 2.449863e-22 3.781559e-23 5.629156e-24 8.080886e-25
##
## 1.118711e-25 1.493552e-26 1.922940e-27 2.387562e-28 2.858820e-29
##
## 3.301126e-30 3.676045e-31 3.947688e-32 4.088350e-33 4.083162e-34
##
## 3.932678e-35 3.652780e-36 3.271914e-37 2.826335e-38 2.354445e-39
##
## 1.891458e-40 1.465374e-41 1.094821e-42 7.888261e-44 5.481036e-45
## step.size
## 3.672716e-46 9.910374e-07- plot the z curve against y
plot(y,z)
D. Vector operation
- x–> -3, 6, 9 | p–> 1/6, 1/2, 1/3
- create two arrays as “x” and “p”
x <- c(-3, 6, 9)
x## [1] -3 6 9p <- c(1/6, 1/2, 1/3)
p## [1] 0.1666667 0.5000000 0.3333333- E(x) = sum(x*p); x1p1 + x2p2 +x3p3 +…
prod(sum(x*p))## [1] 5.5x%*%p## [,1]
## [1,] 5.5sum(x*p)## [1] 5.5- E(x^2)
sum((x*x)*(p*p))## [1] 18.25- E((2x+1)^2)
sum ((2*x) + 1)^2## [1] 729- V(x) = E[(x-E(x))^2] = E[(x-mean)^2] where mean (mu) = E(x) –> E[(x-E(x))^2] * E(x)=sum(x*p)
var(x)## [1] 39- sd(x)
sd(x)## [1] 6.244998- V(2x+1)
var((2*x)+1)## [1] 156E. Poisson distribution
- let x denote a random variable that has a poisson dsitributin with a mean = 4. find the following probabilities
x ~ Possion (lambda) pmf = c^(- lambda) * (lambda^x/x!), x = 0, 1, 2, 3, …
- P(x=5)
dpois(x, lambda)
dpois(5, 4)## [1] 0.1562935- P(x<5)
sum(dpois(0:4, 4))## [1] 0.6288369- P(x>=5)
1-sum(dpois(0:4, 4))## [1] 0.3711631