continuous random variables

Question 1

What is a continuous random variable?

Answer 1

A continuous random variable is one where the possible outcomes or sample space are infinite and uncountable. It can take any value within a range or interval.

Question 2

What is a statistical distribution?

Answer 2

A statistical distribution shows how the values of a variable are arranged, indicating their observed or theoretical frequency of occurrence.

Question 3

What is the probability density function (PDF) of a continuous random variable 𝑋?

Answer 3

The PDF, denoted as f(x), is a function such that for any two numbers 𝑎 and 𝑏, where a≤b, the probability that 𝑋 lies between 𝑎 and 𝑏 is given by:
P(a≤X≤b)=∫f(x)dx (with limits a & b)

Question 4

What are the two conditions that the PDF f(x) must satisfy?

Answer 4

f(x)≥0 for all 𝑥 (the PDF cannot be negative).

The total area under the curve of f(x) from −∞ to +∞ must equal 1

Question 5

what do the two conditions mean in practice

Answer 5

Values of 𝑓(𝑥) cannot be negative
Area under the curve between all possible values of 𝑥 equals , indicating that the probability distribution covers the entire sample space.

Question 6

What is the cumulative distribution function (CDF) for a continuous random variable 𝑋?

Answer 6

The CDF, denoted as F(x), is the probability that 𝑋 takes a value less than or equal to 𝑥:
F(x)=P(X≤x)
It represents the area under the PDF curve up to 𝑥.

Question 7

How do you calculate P(4≤X≤8) using the CDF?

Answer 7

Use the difference of CDF values:
P(4≤X≤8)=P(X≤8)−P(X≤4)

Question 8

What is the formula for the expectation (mean) E(X) of a continuous random variable?

Answer 8

E(X)=∫xf(x)dx (with limits ∞ and -∞)
It represents the weighted average of all possible values of 𝑋.

Question 9

What is the formula for the variance Var(X) of a continuous random variable?

Answer 9

Var(X)=∫(x−E(X))^2f(x)dx (with limit ∞ and -∞)
It measures the spread of the values around the mean.

Question 10

What are the three key parameters that define a theoretical distribution?

Answer 10

Location – Determines where the distribution is centered on the 𝑥-axis.

Scale – Determines the spread of the distribution along the 𝑥-axis.

Shape – Defines the overall form of the distribution (e.g., symmetry, skewness, or tail weight)

Question 11

How is a normal distribution represented mathematically?

Answer 11

X∼N(μ,σ^2)

μ=E(X) (mean, center of the distribution)

σ^2=Var(X) (variance, measures spread)

Question 12

What are three important properties of the normal distribution?

Answer 12

Symmetric about the mean 𝜇, meaning mean = median = mode.

area under the curve area is known (use R)

Bell-shaped with peak at 𝜇.

Question 13

In a normal distribution, what percentage of data falls within one standard deviation of the mean?

Answer 13

68% of data falls within μ±1σ

Question 14

In a normal distribution, what percentage of data falls within two standard deviations of the mean?

Answer 14

95% of data falls within μ±1.96σ

Question 15

In a normal distribution, what percentage of data falls within three standard deviations of the mean?

Answer 15

99% of data falls within μ±2.58σ

Question 16

What is the standard normal distribution?

Answer 16

A normal distribution where μ = 0 and σ² = 1

Question 17

What formula is used to convert a normal variable 𝑋 to a standard normal variable
𝑍?

Answer 17

Z= (X−μ) / σ

Question 18

If Y=aX+b, how does expectation transform?

Answer 18

E(Y)=aE(X)+b

Question 19

If Y=aX+b, how does variance transform?

Answer 19

Var(Y)=a^2Var(X)

Question 20

If height X∼N(63,25) inches, then in cm (Y=2.54X) find expected value and variance

Answer 20

E(Y)=2.54×63=160.02 cm

Var(Y)=2.54^2×25=161.29, so SD(Y)=12.7 cm

Question 21

Suppose that the amounts of money in £, 𝑋, changed at a currency exchange follow a 𝑁(100,𝑠^2=25)
distribution and a flat fee of £2 is charged per transaction. What is expected value and SD of 𝑋 returned in Euros? Assume £1=1.12 Euros

Answer 21

E(Y)=1.12(100)−2.24=112−2.24=109.76

Var(Y)=1.2544×25=31.36

Question 22

What happens when adding or subtracting two independent random variables 𝑋 and 𝑌?

Answer 22

E(X+Y)=E(X)+E(Y)
E(X−Y)=E(X)−E(Y)

Var(X+Y)=Var(X)+Var(Y)
Var(X−Y)=Var(X)+Var(Y)

Question 23

How does covariance affect variance calculations?

Answer 23

If 𝑋 and 𝑌 are not independent, include covariance:

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)

Var(X−Y)=Var(X)+Var(Y)−2Cov(X,Y)

Question 24

What is covariance?

Answer 24

A measure of the joint variability between two random variables 𝑋 and 𝑌

𝐶𝑜𝑣(𝑋,𝑌)=(∑𝑛𝑖(𝑥𝑖−𝑥¯)(𝑦𝑖−𝑦¯)) / 𝑛−1

Question 25

What happens to the sum 𝑋+𝑌 when 𝑋 and 𝑌 are small or large?

Answer 25

Small 𝑋 and 𝑌 → Sum is small
Large 𝑋 and 𝑌 → Sum is large

More variability than uncorrelated variables

Question 26

What happens to the difference 𝑋+𝑌 when 𝑋 and 𝑌 are small or large?

Answer 26

Small: 𝑋 and 𝑌 → Sum is small

large: 𝑋 and 𝑌 → Sum is large

More variability than uncorrelated variables

Question 27

What happens to the difference 𝑋−𝑌 when 𝑋 and 𝑌 are small or large?

Answer 27

Small 𝑋 and 𝑌 → Difference is small

Large 𝑋 and 𝑌 → Difference is small

Less variability than uncorrelated variables

Question 28

What does the correlation coefficient 𝜌 measure?

Answer 28

The strength of the linear relationship between two variables.

Question 29

What is the formula for the correlation coefficient 𝜌?

Answer 29

ρ= Cov(X,Y) / sqrt(Var(X)Var(Y))

Question 30

What are the possible values of 𝜌 and what do they mean?

Answer 30

ρ=−1 → Perfect negative linear relationship

ρ=0 → No relationship

ρ=1 → Perfect positive linear relationship

Question 31

What is the expected value of a linear transformation
Z=a+bX+cY?

Answer 31

E(Z)=a+bE(X)+cE(Y)

Question 32

What is the variance of Z=a+bX+cY if 𝑋 and 𝑌 are independent?

Answer 32

Var(Z)=b^2 Var(X)+c^2Var(Y)