Discrete vs Continuous Distributions

Show your work for each problem.

1.
Which variable is discrete?
1. a)Number of students in a class
2. b)Height of a tree
3. c)Temperature of water
4. d)Distance traveled
Answer: Number of students in a class
Number of students is discrete because you count whole students (15, 16, 17...). Height, temperature, and distance are measured and can take any value in a range, making them continuous.
2.
Which variable is continuous?
1. a)Number of cars in a parking lot
2. b)Number of emails received
3. c)Number of goals scored
4. d)Weight of a newborn baby
Answer: Weight of a newborn baby
Weight is continuous because it's measured and can be 3.245 kg, 3.246 kg, etc. Goals, emails, and cars are counted in whole numbers, making them discrete.
3.
A fair die is rolled. What is ? Give your answer as a fraction.
Answer: 1/6
There are 6 possible outcomes (1, 2, 3, 4, 5, 6) and only one equals 3. So .
4.
Is 'number of pages in a book' discrete or continuous? Answer with one word.
Answer: discrete
Number of pages is discrete because you count whole pages: 148, 149, 150... You cannot have 149.7 pages in a book.
5.
For a continuous distribution, what is ?
1. a)Cannot be determined
2. b)1
3. c)0.5
4. d)0
Answer: 0
For continuous distributions, there are infinitely many possible values. Assigning positive probability to each exact value would make the total exceed 1. Therefore, .
6.
Classify each variable: (a) Time to run a marathon, (b) Number of children in a family, (c) Blood pressure reading
Answer: continuous, discrete, continuous
1. Can marathon time be 3:42:17.38...? Is it discrete or continuous? continuous
2. Can you have 2.5 children? Is number of children discrete or continuous? discrete
3. Can blood pressure be 120.47 mmHg? Is it discrete or continuous? continuous
7.
A fair die is rolled. What is ? Give your answer as a fraction.
Answer: 2/3
Outcomes ≤ 4 are: 1, 2, 3, 4 (4 outcomes). Total outcomes: 6. So .
8.
Why do we use PMF for discrete distributions and PDF for continuous distributions?
1. a)PMF is for large numbers; PDF is for small numbers
2. b)PMF uses decimals; PDF uses fractions
3. c)They are interchangeable terms
4. d)PMF gives exact probabilities; PDF gives density that must be integrated
Answer: PMF gives exact probabilities; PDF gives density that must be integrated
PMF (Probability Mass Function) outputs actual probabilities: . PDF (Probability Density Function) outputs density - you must integrate over an interval to get probability: .
9.
A coin is flipped 3 times. Find where X is the number of heads.
Answer: 3/8
1. How many total outcomes are there when flipping a coin 3 times? 8
2. List outcomes with exactly 2 heads: HHT, HTH, THH. How many? 3
3. What is as a fraction? 3/8
10.
Is 'exact time you wake up' (to the millisecond) discrete or continuous?
Answer: continuous
Time is continuous because it can be measured to any precision. You could wake up at 7:23:45.123456789... seconds. There are infinitely many possible exact times.
11.
Heights follow a normal (continuous) distribution. Explain why , yet people with height 175 cm exist.
Answer: Point probability is 0 but interval probability is positive
1. For continuous distributions, how many exact values can the variable take? infinite
2. If we assigned positive probability to each exact value, what would the sum be? infinite
3. What probability do we use in practice instead of P(X = 175)? P(174.5 < X < 175.5)
12.
A variable can take values 0, 0.5, 1, 1.5, 2, 2.5, ... Is it discrete or continuous?
1. a)Continuous - the values include decimals
2. b)Discrete - the values are listable and separate
3. c)Neither - it's a mixed distribution
4. d)Cannot be determined
Answer: Discrete - the values are listable and separate
This is DISCRETE. The key is that values are separate and listable (0, 0.5, 1, 1.5...), not that they're whole numbers. Between 1 and 1.5, the value cannot be 1.3. Having decimals doesn't make a variable continuous - being able to take ANY value in a range does.
13.
A spinner has 5 equal sections numbered 1-5. Find the probability of getting an even number.
Answer: 2/5
1. Is this a discrete or continuous distribution? discrete
2. What are the even numbers from 1 to 5? 2, 4
3. What is P(even number) as a fraction? 2/5
14.
For a fair die, what is ? Give your answer as a fraction.
Answer: 1/3
Outcomes > 4 are: 5 and 6 (2 outcomes). Total outcomes: 6. .
15.
Is 'amount of rainfall in a month' discrete or continuous? Then, is 'number of rainy days in a month' discrete or continuous?
Answer: continuous, discrete
1. Is rainfall amount (e.g., 45.7 mm) counted or measured? measured
2. So is 'amount of rainfall' discrete or continuous? continuous
3. Is 'number of rainy days' (e.g., 12 days) counted or measured? counted
4. So is 'number of rainy days' discrete or continuous? discrete