WEEK 6

Question 1

What is a continuous random variable?

Answer 1

A variable that can take any value within an interval (e.g., height, temperature). Unlike discrete variables, continuous values are measured, not counted.

Question 2

Key difference between discrete and continuous probability?

Answer 2

Discrete: specific values; Continuous: infinite possible values within a range.

Question 3

What is a Probability Density Function (PDF)?

Answer 3

A function that describes the relative likelihood of a continuous variable. Probability is found as an area under the curve, not a specific point.

Question 4

Why is P(X = x) = 0 for continuous variables?

Answer 4

Because a single point has zero width, meaning zero probability. Probability only applies to intervals.

Question 5

What are the two key properties of a PDF?

Answer 5

1) f(x) ≥ 0 (never negative) 2) The total area under the curve = 1 (100% probability).

Question 6

Formula for calculating probability in continuous distributions?

Answer 6

P(a ≤ X ≤ b) = ∫[a,b] f(x) dx

Question 7

Define the Uniform Distribution.

Answer 7

A distribution where all values in an interval are equally likely. The PDF is flat and constant.

Question 8

Uniform Distribution PDF (not CDF) formula?

Answer 8

f(x) = 1 / (b - a), a ≤ x ≤ b

Question 9

Uniform Distribution Mean Formula?

Answer 9

E(X) = (a + b) / 2

Question 10

Uniform Distribution Variance Formula?

Answer 10

Var(X) = (b - a)^2 / 12

Question 11

Example: A robot restarts every 20 minutes. If you arrive at a random time, what is the probability you wait less than 5 minutes?

Answer 11

P(X < 5) = (5 - 0) / (20 - 0) = 0.25 (Wait time is uniformly distributed U(0,20)).

P(X<x) = (x-a)/ (b-a)

X = random variable
a = minimum possible value of x
b = the maximm possible value of X
x - the time threshold were interested in

Question 12

What is the Normal Distribution?

Answer 12

A bell-shaped, symmetric distribution, defined by mean μ and standard deviation σ.

Question 13

Normal Distribution PDF Formula?

Answer 13

f(x) = (1 / (σ sqrt(2π))) * e^(-(x - μ)^2 / (2σ^2))

Question 14

Why is the Normal Distribution important?

Answer 14

Many natural phenomena follow it, and the Central Limit Theorem states that large sample means tend to be normally distributed.

Question 15

Key properties of a Normal Distribution?

Answer 15

Symmetric, bell-shaped, unbounded, and defined by μ (mean) and σ (spread).

Question 16

What is the Empirical Rule (68-95-99.7 Rule)?

Answer 16

68% of values are within 1σ of the mean. 95% are within 2σ. 99.7% are within 3σ.

Question 17

Example: SAT scores follow N(500, 100). What percentage of students score between 400 and 600?

Answer 17

400 and 600 are 1 standard deviation away, so 68% of scores are in this range (Empirical Rule).

Question 18

What is a Z-score?

Answer 18

A measure of how many standard deviations a value is from the mean.

Question 19

Z-score Formula?

Answer 19

Z = (X - μ) / σ

Question 20

What does a negative Z-score mean?

Answer 20

The value is below the mean. Example: Z = -1.5 means 1.5 standard deviations below the mean.

Question 21

What does a positive Z-score mean?

Answer 21

The value is above the mean. Example: Z = 2 means 2 standard deviations above the mean.

Question 22

Example: If μ = 70, σ = 5, find the Z-score for X = 80.

Answer 22

Z = (80 - 70) / 5 = 2

Question 23

How do we use a Z-table?

Answer 23

Find the area (probability) under the normal curve to the left of the Z-score.

Question 24

Example: SAT scores follow N(500, 100). Find P(X > 600).

Answer 24

Z = (600 - 500) / 100 = 1, then lookup P(Z < 1) = 0.8413, so P(X > 600) = 1 - 0.8413 = 0.1587 (15.87%).

Question 25

What does 'Standard Normal Distribution' mean?

Answer 25

A normal distribution with μ = 0, σ = 1 (converted using Z-scores).

Question 26

What is the Exponential Distribution?

Answer 26

A continuous distribution used to model waiting times between events (e.g., time between phone calls).

Question 27

Exponential Distribution PDF Formula?

Answer 27

f(x) = λ e^(-λx), x ≥ 0

Question 28

Mean and Standard Deviation of an Exponential Distribution?

Answer 28

E(X) = 1 / λ, σ = 1 / λ

Question 29

Example: If calls arrive at a rate of 2 per minute, what is the expected time between calls?

Answer 29

E(X) = 1 / 2 = 0.5 minutes

Question 30

What is the relationship between Poisson and Exponential distributions?

Answer 30

If event arrivals follow a Poisson(λ) process, then the time between events follows Exponential(λ).

Question 31

What is the Normal Approximation to the Binomial?

Answer 31

If n is large and p is not too close to 0 or 1, the Binomial(n, p) can be approximated by a Normal distribution.

Question 32

When can we use the Normal Approximation to Binomial?

Answer 32

If np ≥ 10 and nq ≥ 10 (Success/Failure Rule).

Question 33

Formula for Normal Approximation to Binomial?

Answer 33

X ~ N(np, sqrt(npq))

Question 34

Why do we use the Continuity Correction when approximating Binomial with Normal?

Answer 34

Since the binomial is discrete, and normal is continuous, we adjust by 0.5 when finding probabilities.

Question 35

Example: A factory rejects 7% of screens. If 500 are made daily, what is the expected number rejected?

Answer 35

E(X) = np = 500(0.07) = 35

Question 36

Example: If GMAT scores follow N(500, 100), what score is needed to be in the top 10%?

Answer 36

Find Z = 1.28 (90th percentile). Solve for X: X = μ + Zσ = 500 + (1.28)(100) = 628

Question 37

What is a Normal Probability Plot?

Answer 37

A graph used to check if a dataset follows a normal distribution. If the points form a straight line, data is normal.

Question 38

What is the Central Limit Theorem (CLT)?

Answer 38

The sampling distribution of the mean becomes normal as the sample size increases, regardless of the population’s distribution.

Question 39

Key takeaway of Continuous Probability Models?

Answer 39

Continuous probabilities require integration, and only intervals have probability. Many real-world processes follow Normal, Uniform, or Exponential distributions.