Discrete probability distributions

Question 1

What does
P(X=x) represent ?

Answer 1

The probability of the random variable X taking the value x.

Question 2

What is a discrete random variable (DRV) ?

Answer 2

A discrete random variable (DRV) can only take certain values within a set. It is typically associated with counting something.

Question 3

How do you calculate probabilities using a discrete probability distribution ?

Answer 3

Draw a table to represent the probability distribution.

If it’s given as a function, find each probability.

If any probabilities are unknown, use algebra to represent them.

Ensure the sum of all probabilities equals 1:
∑P(X=x)=1

Question 4

What is a binomial distribution ?

Answer 4

A binomial distribution is a discrete probability distribution that counts the number of successes in a fixed number of trials, where:

There is a fixed finite number of trials (n).

Each trial is independent of others.

There are exactly two possible outcomes (success or failure).

The probability of success (p) is constant.

Question 5

How is a binomial distribution denoted ?

Answer 5

If 𝑋 follows a binomial distribution, it is denoted as:
X∼B(n,p)

Where:
𝑛 is the number of trials.

𝑝 is the probability of success on each trial.

Question 6

What is the relationship between binomial distribution and binomial expansion ?

Answer 6

The binomial distribution can be linked to the binomial expansion of:
(p+(1−p)) ^n

Question 7

How do you set up a binomial model ?

Answer 7

Identify the trial: What is the experiment or action?

Example: rolling a dice, flipping a coin, checking hair colour.

Identify the successful outcome: What outcome counts as a success?

Example: rolling a 6, landing on tails, having black hair.

Define the random variable: Clearly state what the random variable represents.

Example: Let
𝑋 be the number of students in a class of 30 with black hair.

Question 8

What can be modelled using a binomial distribution ?

Answer 8

Fixed number of trials (𝑛).

Two possible outcomes (success or failure).

Independent trials (outcome of one trial doesn’t affect others).

Constant probability of success (𝑝).

Question 9

An example of a binomial model:

Answer 9

Example: Let
𝑇
T be the number of times a fair coin lands on tails when flipped 20 times:

T∼B(20,0.5)

Trial: Flipping the coin.

Number of trials:
𝑛 = 20.

Success: Landing on tails.

Probability of success:
𝑝 = 0.5.

Independence: Each flip does not affect others.

Question 10

What cannot be modelled using a binomial distribution ?

Answer 10

Number of trials is not fixed or is infinite:

Example: Number of coin flips until it lands on heads.

Outcome of one trial affects another:

Example: Choosing caramels from a bag with fewer caramels after each choice.

More than two possible outcomes:

Example: Shoe size, or a die roll (since there are more than two possible outcomes).

Probability of success changes:

Example: Swimming a lap under a minute. As the swimmer gets tired, the probability decreases.

Question 11

How do I calculate
𝑃(𝑋 = 𝑥) the probability of a single value for a binomial distribution ?

Answer 11

Use the Binomial Probability Distribution (BPD) function on your calculator.

Input the following values:
𝑥 (the number of successes for which you want to find the probability).

𝑛 (the number of trials).

𝑝 (the probability of success).

Question 12

How do I calculate
P(X ≤ x), the cumulative probability for a binomial distribution ?

Answer 12

Use the Binomial Cumulative Distribution (BCD) function on your calculator.

Input:
𝑥 (the value for which you want to find the cumulative probability).

𝑛 (the number of trials).

𝑝 (the probability of success).

Question 13

How do I calculate
P(X ≥ x) ?

Answer 13

To calculate
P(X ≥ x), use the formula:
P(X ≥ x)=1−P(X ≤ x−1)

For example:
P(X ≥ 10)=1−P(X ≤ 9)

Question 14

How do I calculate
P(a ≤ X ≤ b) ?

Answer 14

To calculate the probability that 𝑋 is between 𝑎 and, use the formula:

P(a ≤ X ≤ b)=P(X ≤ b)−P(X ≤ a −1)
For example:

P(4 ≤ X ≤ 9)=P(X ≤ 9)−P(X ≤ 3)

Question 15

What is a normal distribution ?

Answer 15

A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped.
It is denoted by X∼N(μ,σ^2),

where:
𝜇 is the mean
𝜎^2 is the variance
𝜎 is the standard deviation

Question 16

How does the variance affect the graph of a normal distribution ?

Answer 16

Changing the variance
𝜎^2 of a normal distribution stretches or shrinks the graph horizontally.

A small variance results in a tall, narrow curve.

A large variance results in a short, wide curve.

Question 17

What is the relationship between the mean, median, and mode in a normal distribution ?

Answer 17

In a normal distribution, the mean, median, and mode are all equal and are located at the centre of the distribution, 𝜇

Question 18

How many points of inflection does the normal distribution curve have and how do you find them ?

Answer 18

The normal distribution curve has two points of inflection, located at
𝑥 = 𝜇 ± 𝜎,
one standard deviation away from the mean.

Question 19

What percentage of data lies within one standard deviation of the mean in a normal distribution ?

Answer 19

Approximately 68%

Question 20

What percentage of data lies within two standard deviations of the mean in a normal distribution ?

Answer 20

Approximately 95%

Question 21

What percentage of data lies within three standard deviations of the mean in a normal distribution ?

Answer 21

Nearly all of the data (99.7%)

Question 22

What is a z-score in a normal distribution ?

Answer 22

measures how many standard deviations a particular value
𝑥 is away from the mean.
(formula on equation sheet)

Question 23

What type of real-life variables can be modelled by a normal distribution ?

Answer 23

Real-life continuous variables that are symmetrical with one mode and come from a large enough population can be modelled by a normal distribution. Examples include height, weight, and test scores.

Question 24

What kind of variables cannot be modelled by a normal distribution ?

Answer 24

Variables that are not symmetrical or have more than one mode (multimodal) cannot be modelled by a normal distribution. Examples include a random number generator output or the distribution of human lifespans.

Question 25

Why can't variables with multiple modes be modelled by a normal distribution ?

Answer 25

A normal distribution is unimodal (one peak).

Question 26

How do you find probabilities using a normal distribution ?

Answer 26

The area under the normal curve between two points 𝑎 and 𝑏 represents the probability 𝑃(a < X < b). Use the "Normal Cumulative Distribution" function on your calculator to calculate the probabilities.

Question 27

Q: What is the probability of a single value for a normal distribution ?

Answer 27

The probability of a single value is always zero: 𝑃 (X = x) = 0, since the area of a single line is zero in a continuous distribution.

Question 28

How do you calculate 𝑃(a < X < b) for a normal distribution ?

Answer 28

Use the "Normal Cumulative Distribution" function on your calculator.

Question 29

How do you calculate 𝑃(X > a) or 𝑃(X < b) for a normal distribution ?

Answer 29

For 𝑃(X>a), use a very large upper bound (e.g.,99999999 or 1099). For 𝑃(X < b), use a very small lower bound (e.g., -99999999 or -1099). These approximations are accurate when the values are more than 4 standard deviations from the mean.

Question 30

How do you find the value of 𝑎 when P(X < a) = p ?

Answer 30

Use the "Inverse Normal Distribution" function on your calculator

Question 31

How do you find a given 𝑝(X > a) = p ?

Answer 31

calculate 𝑝(X < a) = 1−p and use the "Inverse Normal Distribution" function to find 𝑎

Question 32

What is the standard normal distribution ?

Answer 32

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is denoted by 𝑍∼𝑁(0,1^2)

Question 33

Why is the standard normal distribution important ?

Answer 33

Any normal distribution can be transformed into the standard normal distribution by shifting and stretching. (formula is on equation sheet)

Question 34

What happens to the z-value when a value of 𝑋 is less than the mean ?

Answer 34

The z-value will be negative.

Question 35

What does the table of percentage points of the normal distribution provide ?

Answer 35

The table lists values of 𝑧 corresponding to specific cumulative probabilities (e.g. 𝑝(Z ≤ z) = p) for commonly used probabilities

Question 36

When are the values from the standard normal distribution table useful ?

Answer 36

Finding an unknown mean and/or variance for a normal distribution. Performing a hypothesis test on the mean of a normal distribution.

Question 37

What is the first step when finding an unknown mean (μ) or standard deviation (σ) ?

Answer 37

Sketch the normal curve, label the known value of 𝑥 and the mean 𝜇

Question 38

How do you find the z-value for a given probability ?

Answer 38

Use the Inverse Normal Distribution function

Question 39

What should you do after finding the z-value ?

Answer 39

sub it in to the equation on the formular sheet

Question 40

How do you find the mean (μ) and standard deviation (σ) when both are unknown ?

Answer 40

If both are unknown, you will be given two probabilities for two specific values of 𝑥. Follow the same steps but calculate two z-values. Use the rearranged formula for both values of 𝑥.

Question 41

What do you do after calculating two z-values when both 𝜇 and 𝜎 are unknown ?

Answer 41

Form two equations from the two z-values and solve them simultaneously

Question 42

When should you use a binomial distribution ?

Answer 42

The random variable is discrete. The variable counts something (e.g., number of successful trials). There are four conditions: Fixed number of trials (𝑛). Trials are independent. Exactly two outcomes (success or failure). Probability of success (𝑝) is constant for each trial.

Question 43

When should you use a normal distribution ?

Answer 43

The random variable is continuous (e.g., height, weight). The distribution is symmetrical and bell-shaped. The variable measures something, and the data histogram is roughly symmetrical and bell-shaped. As more data is collected, the histogram becomes smoother and resembles a normal distribution curve

Question 44

Can binomial distribution and normal distribution be used in the same question ?

Answer 44

You may first use the normal distribution to approximate a probability and then use the binomial distribution. This typically occurs when sampling is involved. Ensure you are clear about what each parameter/variable represents in the context of the problem.

Question 45

When can I use a normal distribution to approximate a binomial distribution?

Answer 45

when n (the number of trials) is large. 𝑝 (the probability of success) is close to 0.5. The mean (𝜇) and variance (𝜎^2) of the binomial distribution are: 𝜇 = 𝑛𝑝 𝜎^2= 𝑛𝑝(1−𝑝)