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.
- Bernoulli
- Binomial
- Hypergeometric: Qn 1
- Poisson: Qn 2
- Geometric: Qn 3
- Negative Binomial: Qn 4
- Normal: Qn 5
- Exponential: Qn 6
- Chi-square: Qn 7
- Student’s t: Qn 8
- F: Qn 9 Optional Beta Optional
- 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 experiment—one 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 firstp = 0.2 and the second (in black) has a probability of success p = 0.5.
(in gray) has a probability of success
| x <- 0:1
plot(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,(xi = 0) = 1≠ p. The plot below displays three binomial distributions, all forp = 0.1; and in green, p = 0.9. q i in = 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) pmf”5 , 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 it’s 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”)
Chi−square(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”)
Student’s 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) pdf”F(12, 15) pdf ) |
F(12, 15) pdfF(2, 5) pdf
x
