Probability Distributions

Question 1

If ________ is collected from an experiment it can be modelled by a _____________ ______________. The probability of each outcome determines the shape (or distribution).

Answer 1

data
probability distribution

Question 2

What is the definition of Random Variables, X?

Answer 2

A variable whose outcomes (values) are determined by the result of an experiment or observation. Cannot be predicted in advance.

Question 3

What notation is used in probability distributions for random variables?

Answer 3

Uppercase X, represents the name of a random variable (e.g X=flipping a coin). Lower case 𝑥 represents the values X can take.

Question 4

What are the 3 types of random variables?

Answer 4

1) Descriptive
2) Discrete
3) Continuous

Question 5

What is the definition of a Descriptive random variable?

Answer 5

Usually a word or label, e.g X=colour, breed of sheep etc (not included in this unit of work)

Question 6

What is the definition of a Discrete random variable?

Answer 6

Usually whole numbers. E.g X=number of pets in a household. It is possible to calculate P(X=𝑥) e.g P(X=2) is the probability the number of pets in a household is 2.

Question 7

What is the definition of a Continuous random variable?

Answer 7

Determined by a measurement e.g X=time taken to run 100m. Can take on any value on a scale. Cannot find a particular value e.g P(X=13sec), given that values are rounded and probably recorded as groups, so instead we can find e.g P(X<15 or P(10<X<12) (in seconds)

Question 8

When describing distributions what 3 things do you need to include?

Answer 8

1) The highest & lowest values
2) The mode (most common/peak)
3) The shape

Question 9

What are Parameters?

Answer 9

Data calculations e.g median, mean etc

Question 10

What is the Mean?

Answer 10

Average

Question 11

What is Standard Deviation?

Answer 11

Spread

Question 12

What is the full definition of the Mean?

Answer 12

The expected value of a random variable (It is kind of theoretical AVERAGE of the values).
Expected value=E(X)=mean=μ (greek letter “mew”)

Question 13

What is the notation for the Mean?

Answer 13

• Expected value E(X)
• μ (greek letter “mew”)
• Σ𝑥xP(X=𝑥)

Question 14

Find K:
________________________
|P(X=𝑥) | 0.1 | K | 0.2 | 0.1 | 0.3 | (these should all add to 1)

𝑥 | 1 | 2 | 3 | 4 | 5 |

Answer 14

0.1 + 0.2 + 0.1 + 0.3 = 0.7

1 - 0.7 = 0.3

K = 0.3

Question 15

Find E(X) (The expected value of X):
________________________
|P(X=𝑥) | 0.1 | K | 0.2 | 0.1 | 0.3 |
_________________________
(these should all add to 1)

Answer 15

E(X)=(1x0.1)+(2x0.3)+(3x0.2)+(4x0.1)+(5x0.3)=3.2

E(X)=3.2

Question 16

What is Standard Deviation and Variance?

Answer 16

Measures of spread

Question 17

What is the symbol and notation for Standard Deviation?

Answer 17

σ (sigma)

SD(X)

Question 18

What are the 3 equations used for standard deviation?

Answer 18

σ = SD(X)

=√∑(𝑥-μ)² x P(X=𝑥)

=√E(X²) - [(E(X)]²

Question 18

What is the definition of Standard deviation?

Answer 18

Standard deviation tells us how spread the data is from the mean. (The greater the spread, the higher the SD(X)), from formula sheet.

Question 19

What do you need to keep in mind for Variance?

Answer 19

Variance is found by squaring the standard deviation. It is used purely for it’s link to the standard deviation.

Question 20

What is the symbol and notation for Variation?

Answer 20

VAR(X)
σ²

Question 21

What are the steps for using the Graphics Calculator to find E(X) and SD(X)?

Answer 21

Go to STAT menu

To clear table: F6 then F4 (Above Del A) then F1

To enter n values into table List 1 (0 then EXE then 1 then EXE…) Then enter into List 2

F2 to select Calc

F6 to select Set

1 Var X list: List 1

1 Var Freq: List 2

EXE

F1 = 1 Var (calculation answers)

Question 22

Expectation Algebra: What happens if all values are doubled?

_________________________
y | 2 | 4 | 6
_________________________
P(Y=y) | 3/8 | 1/8 | 4/8 |
_________________________

Answer 22

new E(Y) = 2 x 2.125 (2 x E(X)) = 4.25

new VAR (Y) = 2² x 0.859 (2² x VAR (X))

new SD (Y) = √VAR (Y) = 1.854

Question 23

For combination events what are the 2 equations?

Answer 23

1) e (a X + b Y) = a E(X) + b E(Y)
2) Var (a X + b Y) = a² Var(X) + b² Var(Y)

Question 24

For independent events X and Y are where?

Answer 24

on formula sheet

Question 25

Question: If E(X) = $4.20 Var(X) = $0.80 E(Y) = $3.50 Var(Y) = $1.20 & X and Y are independent find: a) E (X + Y) b) E (2X + 7Y + 1) c) Var (2X + 7Y + 1 d) SD (3X - 2Y - 5)

Answer 25

a) E (X + Y) = $4.20 + $3.50 = $7.70 b) E (2X + 7Y + 1) = 2 x 4.20 + 7 x 3.50 + 1 = $33.90 c) Var (2X + 7Y + 1) = 2² x 0.80 + 7² x 1.20 = $62 d) SD (3X - 2Y - 5) = Step 1 find Var: (3X - 2Y - 5) = 3² x 0.8 - 2² x 1.2 = 2.4 Step 2 use: SD (X) = √Var(X) SD (3X - 2Y - 5) = √2.4 = 1.55 (2dp)

Question 26

Question: E(X) = 1.78 and E(Y) = 2.1 If E(X + Y) = 3.24 are X and Y independent?

Answer 26

If events are independent E(X + Y) = 1.78 + 2.1 = 3.88 However E (X + Y) = 3.24 3.24 ≠ 3.88 therefore events are NOT independent.

Question 27

What are the 2 types of data used in distribution models?

Answer 27

Discrete and Continuous

Question 28

What are the 2 types of Discrete data?

Answer 28

1) Poisson 2) Binomial

Question 29

What are the 3 types of Continuous data?

Answer 29

1) Triangular 2) Rectangular 3) Normal

Question 30

What is Rectangular Distribution (a.k.a uniform)>

Answer 30

It is for continuous data. The probability is the same/similar for all intervals. Minimum value & maximum value is given (note: no mode is given).

Question 31

What is ƒ(𝑥)

Answer 31

Probability Density Function

Question 32

What are the 2 parameters for Rectangular Distribution?

Answer 32

a = minimum value b = maximum value

Question 33

What is the formula for rectangular distribution?

Answer 33

ƒ(𝑥) { 1/b-a for a≤𝑥≤b 0 elsewhere

Question 34

Question: What is the probability a person waits for fewer than 12.5 mins on a Monday? (fewer than 12.5 mins = between 0 and 12.5 min) Note: a=min time=0 b=max time=20 c=lower bound of desired interval=0 d=upper bound of desired interval=12.5 Formula to find probability = d-c / b-a for rectangular

Answer 34

P(X<12.5) = 12.5 - 0 / 20 - 0 = 0.625

Question 35

What is Triangular Distribution?

Answer 35

For continuous data. Min and Max AND a Mode is given. Parameters: a=lowest value b=highest value c=mode (peak)

Question 36

How do you find peak height in Triangular Distribution?

Answer 36

2 / b-a = h

Question 37

Where are the equations for triangular distribution?

Answer 37

On the formula sheet (too hard to put on here)

Question 38

What is Normal Distribution?

Answer 38

Used for continuous data. No UPPER OR LOWER limit to the values it can take. The distribution is BELL-SHAPED and SYMMETRICAL

Question 39

What are the parameters for Normal Distribution?

Answer 39

μ = Mean (middle-cos of symmetry) σ = standard deviation (spread) Ƶ = the number of standard deviations a value "𝑥" is from the mean (to the left or right).

Question 40

What are Ƶ values?

Answer 40

Our 𝑥 values are not always exactly 1,2 or 3σ from the mean. Ƶ VALUES tell us HOW MANY STANDARD DEVIATIONS an 𝑥 value is from the mean.

Question 41

What is the formula for calculating a Ƶ value?

Answer 41

Ƶ = 𝑥 - μ / σ

Question 42

What are the steps to finding the Ƶ value?

Answer 42

1) Plug numbers into formula 2) calculate 3) round to (3dp) 4) Look up the number in the table on formula sheet 5) That's your answer

Question 43

What is Inverse Normal Distribution?

Answer 43

Normal Distribution in reverse. Recall: Ƶ = 𝑥 - μ / σ In these questions you will be given 2 of 𝑥, μ or σ as well as a probability from which you can use the normal distribution tables in reverse to get the Ƶ value. You will then use the above equation to work out the missing value.

Question 44

What are the 4 steps for Inverse Normal Distribution?

Answer 44

1) Draw graph. 2) Look up the probability in the middle of the table and work back to the Ƶ value. 3) Calculate missing value using Ƶ = 𝑥 - μ / σ 4) Write in context

Question 45

What is Poisson Distribution (Discrete data)?

Answer 45

It is for rare events, therefore take on shape of skew to the right. For discrete data collected in a "finite continuous space" i.e a given time period, or a specified area etc. E.g number of shark attacks at a beach in one year, or number of tornadoes in a district in a season.

Question 46

What is the parameter for Poisson Distribution?

Answer 46

(lambda) λ = the average number of occurrences in a finite continuous space. λ = variance also (Therefore √λ = standard deviation).

Question 47

What are values of λ estimated from?

Answer 47

Real life data. E.g λ = 2 shark attacks a year.

Question 48

What are the steps for Poisson Distribution in the Graphics Calculator?

Answer 48

Menu then Stat then F5 (dist) then F6 (▻) then F1 (Poisn) then F1 (Ppd) or F2 (Pcd)

Question 49

What is Ppd?

Answer 49

just one value

Question 50

What is Pcd

Answer 50

More than one value

Question 51

Question: Quitline receives on average 8 calls per hour. What is the probability that it receives exactly 3 calls per hour? Hint use Ppd

Answer 51

Data : Variable λ : 8 𝑥 : 3 P(X=3) = 0.0286

Question 52

Question: Quitline receives on average 8 calls per hour. What is the probability that it receives less than 5 calls per hour? Hint use Pcd

Answer 52

λ : 8 𝑥 : 4 P(X<5) = 0.0996 (4dp) X<5 is the same as X≤4

Question 53

Question: Quitline receives on average 8 calls per hour. What is the probability that it receives at least 6 calls per hour? Hint use Pcd

Answer 53

λ : 8 at least 6: 6, 7, 8, 9........... 𝑥 : 5 (G.C works out less than or equal to 𝑥) P(X≥6) = 1 - 0.1912 (4dp) = 0.8088 (4dp)

Question 54

What symbol does a graphics calculator use to depict standard form?

Answer 54

G.C uses: 2.0006ᴇ⁻³ Instead of: 2.0006 x10⁻³

Question 55

What are the 4 different conditions required for Poisson?

Answer 55

1) Each occurrence is independent (doesn't influence) of others. 2) Events must not occur simultaneously (not for events over a long period). 3) Events must occur randomly and unpredictably. 4) For a small interval the probability of an event occurring is proportional to the size of the interval.

Question 56

What is the formula you use for an Inverse Poisson Problem?

Answer 56

To find λ, need P(X=0) Then use formula: λ = ln x P(X=0) ln means natural log

Question 57

What is Binomial Distribution?

Answer 57

For discrete data. There are only 2 outcomes, we refer to these as success or failure. E.g effectiveness of a drug → effective or not Rolling a dice and getting a 6 → getting a 6 or not

Question 58

What are the 4 conditions for Binomial Distribution?

Answer 58

1) There is a fixed number of identical trials 2) the trials are independent 3) there are only 2 possible outcomes 4) the probability of success on any trial is constant (doesn't change)

Question 59

What are the parameters for Binomial Distribution?

Answer 59

π or p = probability of success (i.e favourable outcome occurring) n = number of trials μ = mean = np (or nπ) on formula sheet its μ=nπ σ = standard deviation = √np(1-p) or √nπ(1-π) ← that ones what's on formula sheet Therefore variance = np(1-p)

Question 60

Binomial Distribution graphs points to remember:

Answer 60

* Unimodal (one peak) * Could be skewed to the right or left or symmetrical * Fixed number of trials * 2 outcomes * Independent trials * Probability success=constant

Question 61

Steps for using Binomial Distribution on graphics calculator

Answer 61

Menu then Stat then F5 (dist) then F5 (Binm) then F1 (Bpd) or F2 (Bcd) 𝑥 = number of interest numtrial = how many in the trial, n p = probability success of 1 trial

Question 62

What do you need to know for Inverse Binomial Distribution?

Answer 62

Must know P(X=0) (can get from P(𝑥≥ 1) if needed) or P(X=n) e.g if 30 trials then need P(X=30). You need either n, number of trials/items or P/π, the probability to find the other.