For the following probability distributions: 1. Plot the Probability Function (P

Question

For the following probability distributions:

1. Plot the Probability Function (PMF or PDF) and CDF in different panels,

2. Calculate mean, standard deviation, and variance for each distribution and summarize them in a table (Table 1).

3. Calculate 25th, 50th, and 75th percentiles and summarize them in a table (Table 2).

4. Just for the Binomial distribution: Calculate the CDF for all possible values of X, Approximate CDF for all possible values of X using a Normal distribution and compare them with the exact values that you have calculated (Table 3).

**R STUDIO**

Distributions:

1. Binomial(n=10, p = 0.4)

2. Geometric(p = 0.4)

3. Uniform(0, 10)

4. Normal(0, 1)

5. Normal(0, 5)

6. Normal(5, 10)

7. Weibull(shape = 1, scale = 2)

8. Weibull(shape = 2, scale = 5)

9. Gamma(shape = 2, scale = 5)

10. Lognormal(0, 1)

Explanation / Answer

rm(list=ls())
x<-50
#Binomila Distribution:
xgeom<-rgeom(x,10,0.4)
mean<-mean(xbinom)
Variance<-(49/50)*var(xbinom)
SD<-sqrt((49/50)*var(xbinom))
p25<-qbinom(0.25,10,0.4)
p50<-qbinom(0.50,10,0.4)
p75<-qbinom(0.75,10,0.4)

# Geometric Distribution
xgeom<-rgeom(x,0.4)
mean<-mean(xgeom)
Variance<-(49/50)*var(xgeom)
SD<-sqrt((49/50)*var(xgeom))
p25<-qgeom(0.25,0.4)
p50<-qgeom(0.50,0.4)
p75<-qgeom(0.75,0.4)

# Uniform Distribution
xunif<-runif(x,0,10)
mean<-mean(xunif)
Variance<-(49/50)*var(xunif)
SD<-sqrt((49/50)*var(xunif))
p25<-qunif(0.25,0,10)
p50<-qunif(0.50,0,10)
p75<-qunif(0.75,0,10)

# Normal Distribution(0,1)
xnorm<-rnorm(x,0,1)
mean<-mean(xnorm)
Variance<-(49/50)*var(xnorm)
SD<-sqrt((49/50)*var(xnorm))
p25<-qnorm(0.25,0,1)
p50<-qnorm(0.50,0,1)
p75<-qnorm(0.75,0,1)

# Normal Distribution(0,5)
xnorm<-rnorm(x,0,5)
mean<-mean(xnorm)
Variance<-(49/50)*var(xnorm)
SD<-sqrt((49/50)*var(xnorm))
p25<-qnorm(0.25,0,5)
p50<-qnorm(0.50,0,5)
p75<-qnorm(0.75,0,5)

# Normal Distribution(5,10)
xnorm<-rnorm(x,5,10)
mean<-mean(xnorm)
Variance<-(49/50)*var(xnorm)
SD<-sqrt((49/50)*var(xnorm))
p25<-qnorm(0.25,5,10)
p50<-qnorm(0.50,5,10)
p75<-qnorm(0.75,5,10)

# Weibull Distribution(1,2)
xweibull<-rweibull(x,1,2)
mean<-mean(xweibull)
Variance<-(49/50)*var(xweibull)
SD<-sqrt((49/50)*var(xweibull))
p25<-qweibull(0.25,1,2)
p50<-qweibull(0.50,1,2)
p75<-qweibull(0.75,1,2)

# Weibull Distribution(2,5)
xweibull<-rweibull(x,2,5)
mean<-mean(xweibull)
Variance<-(49/50)*var(xweibull)
SD<-sqrt((49/50)*var(xweibull))
p25<-qweibull(0.25,2,5)
p50<-qweibull(0.50,2,5)
p75<-qweibull(0.75,2,5)

# Gamma Distribution(2,5)
xgamma<-r(x,2,5)
mean<-mean(xgamma)
Variance<-(49/50)*var(xgamma)
SD<-sqrt((49/50)*var(xgamma))
p25<-qgamma(0.25,2,5)
p50<-qgamma(0.50,2,5)
p75<-qgamma(0.75,2,5)

mean variance SD p25 p50 p75 Binomial(n=10,p=0.4) 4.10 2.41 1.55 3.00 4.00 5.00 Geometric(p=0.4) 1.86 3.12 1.76 0.00 1.00 2.00 Uniform(0,10) 5.26 7.99 2.82 2.50 5.00 7.50 Normal(0,1) -0.07 0.97 0.99 -0.67 0.00 0.67 Normal(0,5) -0.46 20.71 4.55 -3.37 0.00 3.37 Normal(5,10) 3.49 81.36 9.02 -1.74 5.00 11.74 Weibul(1,2) 1.68 1.75 1.32 0.58 1.38 2.77 Weibul(2,5) 4.45 5.04 2.24 2.68 4.16 5.88 Gamma(2,5) 0.38 0.08 0.27 0.19 0.34 0.54

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