Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Year 13 > Probability Distributions > Flashcards

Probability Distributions Flashcards

1
Q

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

A

data
probability distribution

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What is the definition of Random Variables, X?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What notation is used in probability distributions for random variables?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What are the 3 types of random variables?

A

1) Descriptive
2) Discrete
3) Continuous

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What is the definition of a Descriptive random variable?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is the definition of a Discrete random variable?

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

What is the definition of a Continuous random variable?

A

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)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What are Parameters?

A

Data calculations e.g median, mean etc

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

What is the Mean?

A

Average

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What is Standard Deviation?

A

Spread

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What is the full definition of the Mean?

A

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”)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is the notation for the Mean?

A
  • Expected value E(X)
  • μ (greek letter “mew”)
  • Σ𝑥xP(X=𝑥)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

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 |

A

0.1 + 0.2 + 0.1 + 0.3 = 0.7

1 - 0.7 = 0.3

K = 0.3

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

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)

A

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

E(X)=3.2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What is Standard Deviation and Variance?

A

Measures of spread

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

What is the symbol and notation for Standard Deviation?

A

σ (sigma)

SD(X)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What are the 3 equations used for standard deviation?

A

σ = SD(X)

=√∑(𝑥-μ)² x P(X=𝑥)

=√E(X²) - [(E(X)]²

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What is the definition of Standard deviation?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

What do you need to keep in mind for Variance?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is the symbol and notation for Variation?

A

VAR(X)
σ²

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

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

A

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)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Expectation Algebra: What happens if all values are doubled?

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

For combination events what are the 2 equations?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

For independent events X and Y are where?

A

on formula sheet

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

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)

A

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)

26
Q

Question:
E(X) = 1.78 and E(Y) = 2.1

If E(X + Y) = 3.24 are X and Y independent?

A

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.

27
Q

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

A

Discrete and Continuous

28
Q

What are the 2 types of Discrete data?

A

1) Poisson
2) Binomial

29
Q

What are the 3 types of Continuous data?

A

1) Triangular
2) Rectangular
3) Normal

30
Q

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

A

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).

31
Q

What is ƒ(𝑥)

A

Probability Density Function

32
Q

What are the 2 parameters for Rectangular Distribution?

A

a = minimum value
b = maximum value

33
Q

What is the formula for rectangular distribution?

A

ƒ(𝑥) { 1/b-a for a≤𝑥≤b
0 elsewhere

34
Q

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

A

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

35
Q

What is Triangular Distribution?

A

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

36
Q

How do you find peak height in Triangular Distribution?

A

2 / b-a = h

37
Q

Where are the equations for triangular distribution?

A

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

38
Q

What is Normal Distribution?

A

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

39
Q

What are the parameters for Normal Distribution?

A

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).

40
Q

What are Ƶ values?

A

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.

41
Q

What is the formula for calculating a Ƶ value?

A

Ƶ = 𝑥 - μ / σ

42
Q

What are the steps to finding the Ƶ value?

A

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

43
Q

What is Inverse Normal Distribution?

A

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.

44
Q

What are the 4 steps for Inverse Normal Distribution?

A

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

45
Q

What is Poisson Distribution (Discrete data)?

A

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.

46
Q

What is the parameter for Poisson Distribution?

A

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

47
Q

What are values of λ estimated from?

A

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

48
Q

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

A

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

49
Q

What is Ppd?

A

just one value

50
Q

What is Pcd

A

More than one value

51
Q

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

A

Data : Variable
λ : 8
𝑥 : 3

P(X=3) = 0.0286

52
Q

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

A

λ : 8
𝑥 : 4

P(X<5) = 0.0996 (4dp)

X<5 is the same as X≤4

53
Q

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

A

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)

54
Q

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

A

G.C uses:
2.0006ᴇ⁻³

Instead of:
2.0006 x10⁻³

55
Q

What are the 4 different conditions required for Poisson?

A

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.

56
Q

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

A

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

ln means natural log

57
Q

What is Binomial Distribution?

A

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

58
Q

What are the 4 conditions for Binomial Distribution?

A

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)

59
Q

What are the parameters for Binomial Distribution?

A

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)

60
Q

Binomial Distribution graphs points to remember:

A
  • Unimodal (one peak)
  • Could be skewed to the right or left or symmetrical
  • Fixed number of trials
  • 2 outcomes
  • Independent trials
  • Probability success=constant
61
Q

Steps for using Binomial Distribution on graphics calculator

A

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

62
Q

What do you need to know for Inverse Binomial Distribution?

A

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.

Year 13 (4 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions