gallery of continuous variables

Question 1

Parametrized distributions

Answer 1

When we studied discrete random variables we learned, for example, about the Bernoulli(𝑝) distribution. The probability 𝑝 used to define the distribution is called a parameter and Bernoulli(𝑝) is called a parametrized distribution.

Question 2

what distribution do tosses of a fair coin follow?

Answer 2

tosses of fair coin follow a Bernoulli distribution where the parameter 𝑝 = 0.5.

Question 3

what is a key question of statistics?

Answer 3

estimate the parameters of a distribution

Question 4

give an example of estimating the parameters of a distribution

Answer 4

if I have a coin that may or may not be fair then I know it follows a Bernoulli(𝑝) distribution, but I don’t know the value of the parameter 𝑝.

I might run experiments and use the data to
estimate the value of 𝑝.

As another example, the binomial distribution Binomial(𝑛, 𝑝) depends on two parameters 𝑛 and 𝑝.

Question 5

Uniform Distribution
Parameters, Range, Notation, PDF, CDF, Models

Answer 5

1. Parameters: 𝑎, 𝑏.
2. Range: [𝑎, 𝑏].
3. Notation: uniform(𝑎, 𝑏) or U(𝑎, 𝑏).

Question 6

uniform distribution graphs

Answer 6

Question 7

Example 1. 1. Suppose we have a tape measure with markings at each millimeter. If we
measure (to the nearest marking) the length of items that are roughly a meter long, the
rounding error will be uniformly distributed between -0.5 and 0.5 millimeters.

Answer 7

Many board games use spinning arrows (spinners) to introduce randomness. When spun, the arrow stops at an angle that is uniformly distributed between 0 and 2𝜋 radians.

3. In most pseudo-random number generators, the basic generator simulates a uniform distribution and all other distributions are constructed by transforming the basic generator.

Question 8

exponential distribution

Answer 8

1. Parameter: 𝜆.
2. Range: [0, ∞).
3. Notation: exponential(𝜆) or exp(𝜆).

Question 9

taxi waiting time

Answer 9

If I step out to 77 Mass Ave after class and wait for the next taxi, my waiting time in minutes is exponentially distributed. We will see that in this case 𝜆 is given by 1/(average number of taxis that pass per minute).

Question 10

waiting time for unstable isotope to undergo nuclear decay

Answer 10

The exponential distribution models the waiting time until an unstable isotope
undergoes nuclear decay. In this case, the value of 𝜆 is related to the half-life of the isotope

Question 11

true or false: exponential distribution is the only one that models waiting times

Answer 11

False.

there are other distributions that also model waiting times, but the exponential distribution has the property that it is memoryless.

Question 12

Example of memoryless wait time

Answer 12

Suppose that the probability that a taxi arrives within
the first five minutes is 𝑝. If I wait five minutes and, in this case, no taxi arrives, then the
probability that a taxi arrives within the next five minutes is still 𝑝. That is, my previous
wait of 5 minutes has no impact on the length of my future wait!

Question 13

Example of wait time with memory

Answer 13

Suppose I were to instead go to Kendall Square subway station and wait for the next inbound train. Since the trains are coordinated to follow a schedule (e.g., roughly 12 minutes between trains), if I wait five minutes without seeing a train then there is a far greater probability that a train will arrive in the next five minutes.

In particular, waiting
time for the subway is not memoryless, and a better model would be the uniform distribution on the range [0,12].

Question 14

what is the memorylessness of the exponential distribution analagous to?

Answer 14

The memorylessness of the exponential distribution is analogous to the memorylessness
of the (discrete) geometric distribution, where having flipped 5 tails in a row gives no information about the next 5 flips. Indeed, the exponential distribution is precisely the continuous counterpart of the geometric distribution, which models the waiting time for a discrete process to change state. More formally, memoryless means that the probability of waiting 𝑡 more minutes is independent of the amount of time already waited.

In symbols,

𝑃 (𝑋 > 𝑠 + 𝑡 | 𝑋 > 𝑠) = 𝑃(𝑋 > 𝑡).

Question 15

proof of memorylessness

Answer 15

We know that

(𝑋 > 𝑠 + 𝑡) ∩ (𝑋 > 𝑠) = (𝑋 > 𝑠 + 𝑡),

since the event ‘waited at least 𝑠 minutes’ contains the event ’waited at least 𝑠 + 𝑡 minutes’. Therefore the formula for conditional probability gives

Question 16

exponential distribution graphs

Answer 16

Question 17

Normal Distribution or Gaussian Distribution?

Answer 17

In 1809, Carl Friedrich Gauss published a monograph introducing several notions that have become fundamental to statistics: the normal distribution, maximum likelihood estimation, and the method of least squares (we will cover all three in this course).

For this reason,
the normal distribution is also called the Gaussian distribution, and it is by far the most important continuous distribution.

Question 18

Normal Distribution

Answer 18

Question 19

standard normal distribution

Answer 19

𝑁 (0, 1) has mean 0 and variance 1.

Question 20

Z

Answer 20

standard normal random variable

Question 21

𝜙(𝑧)

Answer 21

standard normal density

Question 22

normal distribution graphs

Answer 22

Question 23

different normal distributions

Answer 23

Question 24

three approximate probabilities for the standard normal distribution

Answer 24

𝑃 (−1 ≤ 𝑍 ≤ 1) ≈ 0.68

𝑃 (−2 ≤ 𝑍 ≤ 2) ≈ 0.95

𝑃 (−3 ≤ 𝑍 ≤ 3) ≈ 0.99

Question 25

describe a figure to name the three approximate probabilities as areas under the graph of the standard normal pdf 𝜙(𝑧).

Answer 25

Question 26

symmetry calculations

Answer 26

We can use the symmetry of the standard normal distribution about 𝑧 = 0 to make some calculations.

Question 27

The rule of thumb says 𝑃 (−1 ≤ 𝑍 ≤ 1) ≈ 0.68. Use this to estimate Φ(1).

Answer 27

Φ(1) = 𝑃 (𝑍 ≤ 1). In the figure, the two tails (in blue) have combined area 1 − 0.68 = 0.32. By symmetry the left tail has area 0.16 (half of 0.32), so 𝑃 (𝑍 ≤ 1) ≈
0.68 + 0.16 = 0.84.

Question 28

Using R to compute the standard normal cdf

Answer 28

Use the R function pnorm(𝑥, 𝜇, 𝜎) to compute 𝐹 (𝑥) for N(𝜇, 𝜎^2)

pnorm(1,0,1)
[1] 0.8413447

pnorm(0,0,1)
[1] 0.5

pnorm(1,0,2)
[1] 0.6914625

pnorm(1,0,1) - pnorm(-1,0,1)
[1] 0.6826895

pnorm(5,0,5) - pnorm(-5,0,5)
[1] 0.6826895

Of course z can be a vector of values
pnorm(c(-3,-2,-1,0,1,2,3),0,1)
[1] 0.001349898 0.022750132 0.158655254 0.500000000 0.841344746 0.977249868 0.998650102

Question 29

Use R to compute 𝑃 (−1.5 ≤ 𝑍 ≤ 2)

Answer 29

This is Φ(2) − Φ(−1.5) = pnorm(2,0,1) - pnorm(-1.5,0,1) = 0.91044

Question 30

Pareto distribution

Answer 30