Directions

Use the attached RMarkdown Probability Distributions.Rmd to create a pdf report summarizing the common distributions discussed in this lesson. In addition, I have done the first few to show you my expectations for your report, and have given you some leading questions to complete. Partial R-code for the following distributions have been given for you.

1. Bernoulli
2. Binomial
3. Hypergeometric: Qn 1
4. Poisson: Qn 2
5. Geometric: Qn 3
6. Negative Binomial: Qn 4
7. Normal: Qn 5
8. Exponential: Qn 6
9. Chi-square: Qn 7
10. Students t: Qn 8
11. F: Qn 9 Optional Beta Optional
12. Logistic Optional

All R-code and output must be clearly shown. Late submission will attract a penalty of 10 points per day after the due date.

If you have any questions, please post them on the lesson discussion board.

1 Discrete Distributions

1.1 Bernoulli

The Bernoulli distribution, named for Jacob Bernoulli, assigns probability to the outcomes of a single Bernoulli experimentone where the only possible outcomes can be thought of as a success or a failure

(e.g., a coin toss). Here, the random variable _x can take on the values 1 (success) with probability _p, or 0

(failure) with probability q = 1 p. The plot below contains the pmf of two Bernoulli distributions. The first_p = 0_.2 and the second (in black) has a probability of success _p = 0_.5.

(in gray) has a probability of success

x <- 0:1plot(x, dbinom(x, 1, 0.2), type = h, ylab = f(x), ylim = c(0, 1), lwd = 8, col = darkgray, main = Bernoulli(0.2))lines(x, dbinom(x, 1, 0.5), type = h, lwd = 2, col = black)legend(0.7, 1, c(Bernoulli(0.2), Bernoulli(0.5)), col = c(darkgray, black), lwd = c(8, 2))

Bernoulli(0.2)

x

The Bernoulli experiment forms the foundation for many of the next discrete distributions.

1.2 Binomial

The binomial distribution applies when we perform _n Bernoulli experiments and are interested in the

total number of successes observed. The outcome here, _y = _x , where _P(_x = 1) = _p and _p = 0_.5; in blue,

Pgray,(x_i = 0) = 1 p. The plot below displays three binomial distributions, all for_p = 0_.1; and in green, _p = 0_.9. q _{i i}n = 10 Bernoulli trials: in

x <- seq(0, 10, 1)plot(x, dbinom(x, 10, 0.5), type = h, ylab = f(x), lwd = 8, col = dark gray, ylim = c(0,0.5), main = Binomial(10, 0.5) pmf) lines(x, dbinom(x, 10, 0.1), type = h, lwd = 2, col = blue)lines(x, dbinom(x, 10, 0.9), type = h, lwd = 2, col = green)legend(3, 0.5, c(Binomial(10,0.1), Binomial(10,0.5),

Binomial(10,0.9)), col = c(blue,

dark gray, green), lwd = c(2, 8,

2))

Binomial(10, 0.5) pmf

x

We can see the shifting of probability from low values for _p = 0_.1 to high values for _p = 0_.9. This makes sense, as it becomes more likely with _p = 0_.9 to observe a success for an individual trial. Thus, in 10 trials, more successes (e.g., 8, 9, or 10) are likely. For _p = 0_.5, the number of successes are likely to be around 5 (e.g., half of the 10 trials).

1.3 Hypergeometric

In the example I have below, I have set the number of balls in the urn to 10, 5 of which are white and 5 of which are black. I have also fixed the number of balls drawn from the urn to 5. Play around with the parameters and describe what you see.

x <- seq(0, 10, 1)plot(x, dhyper(x, 5, 5, 5), type = h, ylab = f(x),lwd = 2, main = Hypergeometric(5,10,5) pmf5 , 5 ,5 )

Hypergeometric(5,10,5) pmf5, 5, 5

x

1.4 Poisson

What happens if you increase ? To 2? To 3?

x <- seq(0, 5, 1)

plot(x, dpois(x, 1), type = h, ylab = f(x), main = Poisson(1) pmf, lwd = 2)

Poisson(1) pmf

x

1.5 Geometric

What happens to the geometric distrbution if you vary _p? Show me a few plots and explain.

x <- seq(0, 20, 1)

plot(x, dgeom(x, 0.2), type = h, ylab = f(x), lwd = 2, main = Geometric(0.2) pmf)

Geometric(0.2) pmf

x

1.6 Negative Binomial

The negative binomial I have below has set _r = 1, so its identical to the geometric above. Play around with r and see how it changes.

x <- seq(0, 20, 1)

plot(x, dnbinom(x, 1, 0.2), type = h, ylab = f(x), lwd = 2, main = Negative Binomial(0.2) pmf)

Negative Binomial(0.2) pmf

x

2 Continuous Distributions

2.1 Exponential

Vary and describe.

x <- seq(0, 10, 0.01)

plot(x, dexp(x, 1), type = l, ylab = f(x), lwd = 2, main = Exponential(1) pdf)

Exponential(1) pdf

x

2.2 Normal

Vary and see how the distribution changes. If you make it too big, you may need to adjust the x-axis by

making the sequence span a wider range thanthe proper limits for x for a given 5 to 5. You can use a trial-and-error approach to determing .

x <- seq(5, 5, 0.01)

plot(x, dnorm(x, 0, 1), type = l, ylab = f(x), main = Normal(0, 1) pdf)

Normal(0, 1) pdf

x

2.3 Chisquare

How do the degrees of freedom change the shape? Plot a few and explain.

x <- seq(0, 20, 0.01)

plot(x, dchisq(x, 6), type = l, ylab = f(x), main = Chi-square(6) pdf)

Chisquare(6) pdf

x

2.4 Students t

How do the degrees of freedom change the shape? Plot a few and explain.

x <- seq(5, 5, 0.01)

plot(x, dt(x, 6), type = l, ylab = f(x), main = Student s t(6) pdf)

Students t(6) pdf

x

2.5 F

How do the degrees of freedom (numerator and/or denominator) change the shape? Plot a few and explain.

x <- seq(0, 6, 0.01)plot(x, df(x, 12, 15), type = l, ylab = f(x),main = F(2, 5) pdfF(12, 15) pdf )

F(12, 15) pdfF(2, 5) pdf

x