Continuous Distributions

Question 1

What are Continuous Distributions?

Answer 1

Distributions that have infinitely many consecutive outcomes

• Their sample space is infinite
• We cannot record the frequency of each distinct value
• we cannot represent these distributions with a table
• we represent them with graphs

Question 2

What is PDF?

Answer 2

Probability Density Function

Question 3

How is PDF denoted?

Answer 3

f(y)

the function depicts the associated probability for every possible value of ‘y’

> = 0

Question 4

What does PDC stand for?

Answer 4

The Probability Distribution Curve

Question 5

What does the PDC show?

Answer 5

The likelihood of each outcome

Question 6

The probability of any individual value from a continuous distribution is equal to what number?

Answer 6

Zero

P(X) = 0

Example: The likelihood of an athlete to run the mile in under 6 minutes is = the the likelihood of running it at 6 min

P(x < 6 min ) = P(x <= 6 min)

Question 7

What is CDF? and how is it denoted?

Answer 7

The Cumulative Distribution Function

F(y)

Question 8

What is an Integral?

Answer 8

a function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.

Question 9

What is the opposite of Integration?

Answer 9

Derivation

PDF -> CDF - Integral

PDF <- CDF - Derivative

Question 10

Graphically, what is the relationship between the PDF and the CDF of a continuous distribution?

The PDF represents the area under the curve of the CDF.

The CDF represents the area under the curve of the PDF.

There is no graphical representation of the relationship between the two.

The CDF and PDF are interchangeable definitions, so they express the same curve and are under the graph.

Answer 10

The CDF represents the area under the curve of the PDF.

Question 11

How is a normal distribution dentoed?

Answer 11

N(mean, variance)

Question 12

What is the expected value for a Normal Distribution?

Answer 12

E(X) = it’s mean

Question 13

How you do calculate variance?

Answer 13

Var(X) = E(X2) - E(X)2

Question 14

What is the 68, 95, 99.7 Law?

Answer 14

For any normally distributed event 68% of all outcomes fall within 1 std dev from mean, 95% fall within 2 std deviations, and 99.7% 3 std dev

Question 15

Which of the following is a characteristic of the graph of a Normal Distribution?

Bell-shaped.

Symmetric.

Thin tails.

All of the above.

Answer 15

All of the above.

Question 16

How do you Standardize a Normal Distribution

Answer 16

We make the mean to equal 0 and std dev to equal 1.

E(X) = 0
Var(X) = 1

Step 1: Subtract mean from every element
Step 2: Divide every element of the new distribution by the std dev

Question 17

What is a transformation?

Answer 17

A way in which we can alter every element of a distribution to get a new distribution with similar characteristics

Question 18

What are the two characteristics of a Standard Normal Distribution?

Mean of 0 and a variance of 1.

Mean of 1 and a variance of 0.

Variance of 0 and mean of 1.

Mean of 0 and variance of -1.

Answer 18

Mean of 0 and a variance of 1.

Question 19

When do we use a Student’s T Distribution?

When we have limited data about a variable, which historically follows a Normal Distribution.

When we have limited data about a variable, regardless of other features of the data set.

When the variable possesses characteristics of a Normal Distribution and we have an excess of 30 observations.

None of the above.

Answer 19

When we have limited data about a variable, which historically follows a Normal Distribution.

Question 20

What is a Chi-squared distribution? (Kai)

Answer 20

Greek letter kai Χ

Chi-square is a statistical test that examines the differences between categorical variables from a random sample in order to determine whether the expected and observed results are well-fitting

Question 21

When is Chi-squared used?

Answer 21

Few events in real life follow this distribution.

Most used in:
- hypothesis testing
- computing confidence intervals

used to determine the goodness of fit of categorical values

Question 22

What is one of the features of the graph of a Chi-Squared Distribution?

Bell-shaped.

Symmetric.

Thin tails.

None of the above.

Answer 22

None of the above.

Question 23

What is the Exponential Distribution?

Answer 23

Exp(λ)

rate parameter = lambda

values that:
- Starts off high
- Initially decreases
- Eventually plateauing

example: youtube video

E(Y) = 1 / lambda
Var(Y) = 1 / λ2

Question 24

What is the most common transformation of an Exponential Distribution?

Answer 24

Transforming it into a normal distribution

Y ~ Exp(λ)
X= ln(Y)

Question 25

Which of the following is correct about the graphs of the Exponential Distribution?

The CDF plateaus near 1 mark and the PDF plateaus around the 0 mark.

The PDF plateaus near 1 mark and the CDF plateaus around the 0 mark.

The curve of the CDF resembles a boomerang with its handles lining up with the X- and the Y- axes.

None of the above.

Answer 25

The CDF plateaus near 1 mark and the PDF plateaus around the 0 mark.

Question 26

What is the Logistic Distribution?

Answer 26

Denoted as: Logistic(mean, scale)

We often encounter these distributions when trying to determine how continuous variable inputs (time, distance, etc) can effect the probability of a binary outcome

used in forecasting competitive sporting events - where there exists only two clear outcomes

Does the average speed of a tennis players serve play a crucial role in the outcome of a match?

Question 27

What is true about the CDF of a Logistic Distribution?

It follows an S-shape and plateaus around the 1 value.

The probability drastically starts to increase once we reach values close to the mean.

The steeper the curve is, the faster it reaches value close to absolute certainty (1).

All of the above.

Answer 27

All of the above.