02 Random Variables

Question 1

What is a random variable?

Answer 1

A numerical measure of the outcomes of a random phenomenon.

Question 2

There are two kinds of random variables. What are they?

Answer 2

Discrete and continuous.

• Discrete random variables can take a finite number of outcomes (eg shoe sizes)
• Continuous random variables can take an infinite number of values (eg heights)

Question 3

What is a probability distribution?

Answer 3

A list or graph showing the probabilities of the different possible values the random variable can take

Question 4

A bar chart showing the probabilities associated with a discrete random variable is called what?

Answer 4

A probability histogram.

Question 5

How does a probability distribution for a continuous random variable represent the probabilities?

Answer 5

As areas under a probability density curve.

Question 6

The mean of a probability distribution is also referred to as the…

Answer 6

Expected value, E(X)

Question 7

How do you calculate the mean of a discrete random variable?

Answer 7

μ_X = Σx_i⋅p_i = x₁⋅p₁ + x₂⋅p₂ + x₃⋅p₃ + … + x_n⋅p_n

Question 8

What does the variance of a probability distribution measure?

Answer 8

How far the data is from the mean.

• The larger the variance, the more data points that are far from the mean.
• The smaller the variance, the more data points that are close to the mean

Question 9

How is the variance of a probability distribution calculated?

Answer 9

Var(X) = σ²_X = Σ(X_i-μ_X)²⋅p_i = (X₁-μ)²⋅p₁ + (X₂-μ)²⋅p₂ + …+ (X_n-μ)²⋅p_n

Question 10

What is the standard deviation?

Answer 10

Standard deviation, σ, is the square root of the variance:

• σ = √Var(X)

Question 11

What is a population?

Answer 11

• Properties of the probability distribution are population properties.
• The population is the total number of people/trees/chocolates/objects…. described by the probability distribution

Question 12

What is a sample?

Answer 12

• The data actually measured or observed
• The sample is the group of people/chocolates/trees whose height/size/age you measure/count

Question 13

What is the Law of Large Numbers?

Answer 13

The mean of many trials approaches the true mean of the distribution as the number of trials increases.

Question 14

μ and σ are the symbols for the population or sample mean and standard deviation?

Answer 14

Population

Question 15

x̄ and s are the symbols for the population or sample mean and standard deviation?

Answer 15

Sample

Question 16

Symbolically, the Law of Large Numbers says what?

Answer 16

x̄ → μ as n → ∞

where n = number of trials

Question 17

If X is a random variable and a, b are constants, how are μ_a+bX and σ²_a+bX related to μ_X and σ²_X?

Answer 17

• μ_a+bX = a + b⋅μX
• σ²_a+bX = a + b²⋅σ²_X

Question 18

If X, Y and Z are random variables such that Z = X+Y, then how are the means and variances of X, Y and Z related?

Answer 18

• μ_Z = μ_X + μ_Y
• σ²_Z = σ²_X + σ²_X if X, Y are independent

Question 19

If X, Y and W are random variables such that W = X-Y, then how are the means and variances of X, Y and W related?

Answer 19

• μ_W = μ_X - μ_Y
• σ²_W = σ²_X + σ²_X if X, Y are independent

Question 20

If an event has only two possible outcomes, what distribution can we use if we want to know the probability of getting k successes out of a total number of n trials?

Answer 20

The binomial distribution.

Question 21

What are the 4 criteria for the use of a binomial model?

Answer 21

1. There are only two possible outcomes of each observation (success or failure)
2. There is a fixed number of observations, n
3. The n observations are independent
4. The probability of success, p, is the same for each observation

Question 22

What are the parameters of a distribution?

Answer 22

Parameters are the numbers that completely characterise a distribution. This means that only they are needed to calculate probabilities; nothing else.

Question 23

What are the parameters of the binomial distribution?

Answer 23

• n* and p; B(n, p)
• n* = the total number of trials
• p* = the probability of obtaining a success

Question 24

What is the formula for calculating a probability with the binomial distribution?

Answer 24

The probability that there are k successes in n trials is given by P(X=k) = _nC_k⋅p^k⋅(1-p)^n-k

Question 25

What is the mean and standard deviation of a binomial distribution?

Answer 25

• μ = np
• σ = √np(1-p)

Question 26

What are the 4 features of a geometric model?

Answer 26

1. Each observation can be one of only two outcomes - success or failure
2. Each observation is independent
3. The probability of success, p, is the same for each observation
4. We want to find the probability that we need n trials in order to obtain the first success

Question 27

What are the parameters of a geometric distribution?

Answer 27

p, the probability of success

Question 28

How do you calculate a probability with the geometric distribution?

Answer 28

The probability that the first success is on trial number n is P(X=n) = (1-p)^n-1p

Question 29

What is the expected value of a geometric random variable?

Answer 29

E(X) = μ_X = 1/p

Question 30

What is the variance and standard deviation of a geometric random variable?

Answer 30

σ²_X = 1-p/p² σ²_X = √(1-p)/p

Question 31

What are the 4 key things to know about random variables?

Answer 31

1. a random variable is a numerical measure of the outcome of a random phenomenon
2. if X is a r.v. and a,b are fixed numbers, then
   E(a+bX) = a+b⋅E(X)
   Var(a+bX) = b²⋅Var(X)
3. if X and Y are r.v.’s, then
   E(X±Y) = E(X)±E(Y)
4. if X and Y are independent r.v.’s, then
   Var(X+Y) = Var(X-Y) = Var(X)+Var(Y)