Please find the Standard Deviation of each option

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?

Describe the point estimate, normal probability distribution, and standard normal probability distribution in details, in 4 paragraphs. 

1. Generate 7 integers with equal probability from a function which returns 1/10 with probability P and (1-P)
2. What are the ROC curve and meaning of sensitivity, specificity, confusion matrix
3. What is the definition of a P-value? How to explain p-value to customers

Bayesian probability is subjective insofar as we can start with almost any prior probability we want. Why are we then able to say that Bayesian probability can give us an accurate understanding of what the probability of an event is?

Exercise 1: probability Concepts

Part one: Contingency table and Probability rules

A study on speeding violation and driver age produced the following contingency table.

1. Does it appear that being 60 years of age or more and speed violating are mutually exclusive events? Explain your reasoning

If a driver is selected at random from this sample, find the following probabilities:

Round your answers to 3 decimals

2. The driver had a speeding violation in the last year
3. The driver age is less than 50 years
4. The driver is aged between 30 to 39 and had speeding violation last year
5. The driver had no speeding violation last year or is aged between 60 to 69 years
6. The driver had a speeding violation if he is aged between 20 to 29 years
7. The driver had a speeding violation if he is more than 49 years of age.
8. Does it appear that speed violation is independent from age group for this sample of drivers? Explain your reasoning

Part two: Classical probabilities, counting rules and probabilities

1. A fair coin and then a die with 6 sides are tosses find the probabilities of the six events occurring respectively

1. P(Tails)          
2. P(3)                               
3. P(tails and 3)                
4. P(tails | 3)                  
5. P(3 | tails)                       
6. P(4 or a 5)                      

2. An urn contains 5 white ,4 black and 3 red marbles. If 3 marbles are selected from this urn.

Find the probability that at least one of the 3 marbles is black

Suppose that coin 1 has probability 0.8 of coming up heads, and coin 2 has probability 0.6 of coming up heads. If the coin flipped today comes up heads, then with equal probability we select coin 1 or coin 2 to flip tomorrow, and if it comes up tails, then we select coin 2 to flip tomorrow.

(a) If the coin initially flipped is coin 1, what is the probability that the coin flipped on the second day after the initial flip is coin 2?

(b) What proportion of flips use coin 1 and what proportion use 2 in the long run?

1. Define empirical and theoretical probability.
2. Describe a situation where empirical probability would be used.
3. Explain why your situation represents empirical probability.