Question

1.Plotting densitiesPlot the probability mass function (pmf) or probability density function (pdf) for eachof the following scenarios:(a) Consider abinomialrandom variable,X.i. Plot the pmf ofX∼Bin(n= 10,p= 0.3).ii. Plot the pmf ofX∼Bin(n= 10,p= 0.7).iii. Plot the pmf ofX∼Bin(n= 100,p= 0.3).iv. What happens to the shape of the pmf ofX∼Bin(n,p) whenpgets larger?v. What happens whenngets larger?(b) Consider ageometricrandom variable,Y.i. Plot the pmf ofY∼Geom(p= 0.1).ii. Plot the pmf ofY∼Geom(p= 0.5).iii. Plot the pmf ofY∼Geom(p= 0.8).iv. What happens to the shape of the pmf ofY∼Geom(p) whenpgets larger?(c) Consider aexponentialrandom variable,T.i. Plot the pdf ofT∼Exp(λ= 0.1).ii. Plot the pdf ofT∼Exp(λ= 0.5).iii. Plot the pdf ofT∼Exp(λ= 2).iv. What happens to the shape of the pdf ofT∼Exp(λ) whenλgets larger?(d) Consider anormalrandom variable,M.i. Plot the pdf ofM∼N(μ= 2,σ2= 1).ii. Plot the pdf ofM∼N(μ=−1,σ2= 1).iii. Plot the pdf ofM∼N(μ= 2,σ2= 5).1 iv. What happens to the pdf ofM∼N(μ,σ2) whenμis changed?v. What happens to the pdf ofM∼N(μ,σ2) whenσ2gets larger?(e) Which of the continuous distributions looks the most similar to the geometricdistribution? Which looks the most similar to the binomial distribution (withlargen)? Do these relationships make sense, based on your knowledge of thedistributions and their assumptions?

Answer 1

a) X~B(n,p)

i. B(n=10,p=0.3)

ii. B(n=10,p=0.7)

iii. B(n=100,p=0.3)

iv. We observe that if p goes from 0.3 to 0.7, nature of distribution goes from positively skewed to negatively skewed.

v. When n large, the distribution tends to symmetric.

b) Y~Geometric(p)

i. Geom(p=0.1)

ii. Geom(p=0.5)

iii. Geom(p=0.8)

iv. When p gets larger, the nature of distribution is exponential.

c) T∼Exp(λ)

i. Exp(λ= 0.1)

ii. Exp(λ= 0.5)

iii. Exp(λ= 2)

iv.when λ gets larger then peakness of curve is goes down.

d) M∼N(μ,σ2)

i. N(μ= 2,σ2= 1)

ii. N(μ= -1,σ2= 1)

iii. N(μ= 2,σ2= 5)

iv.When mu is changed the location of the distribution or mean of the distribution changed.

v. When sigma is increasing then peakness of the distribution is goes down, means the large sigma distribution is flatter than small sigma.

e)Exponential distribution looks the most similar to the geometric distribution from the graph.

For large n, the binomial distribution approaches to the normal distribution. Hence normal distribution is similar to look binomial distribution for large n.

Yes, the theoretical knowledge of distribution and their assumptions get satisfied through graphically.