Flashcards in S2 Deck (37):

1

## sum of all the probabilities

### 1

2

## mean for discrete random variables

###
E(X)

sum of (x values times their probabilties added together)

3

## variance for discrete random variables

###
E(X^2) - (E(X))^2

sum of (x values squared times their probabilities added together) - mean squared

4

## standard deviation

### square root of variance

5

## E(g(X))

###
sum of g(xi)pi

only times x not probabilities

6

## E(aX)

### aE(X)

7

## E(X+b)

### E(X)+b

8

## E(aX+b)

### aE(X)+b

9

## Var(aX)

### a^2Var(X)

10

## Var(X+b)

### Var(X)

11

## Var(aX+b)

### a^2Var(X)

12

## conditions of a poisson distribution

###
each event is independent of the other

each event is randomly occuring

two events can't occur at the exact same time

probability of each event occuring is fixed

13

## how to express poisson distribution

### X squiggly line Po(mean)

14

## mean of 2 calls per minute

### = mean of 8 calls in 4 minutes

15

## 20 lorries and 40 cars per minute

### mean of 60 vehicles per minute (make sure theyre in the same unit time before adding)

16

## to be a poisson distribution...

### mean approximately equal to variance

17

##
probability density function

f(x)

###
area under the graph represents the probability

so total area under curve = 1

integrate with bounds to find area/probability

18

##
cumulative distribution function

F(x)

###
probability that value is less than number

19

## convert from pdf to cdf

###
integrate the different sections of the pdf between their limits

always use c as the upper limit to get an answer in terms of c

e.g between 0 and 3, then 3 and 5 to find cdf between 3 and 5 integrate pdf between these limits with upper limit c instead of 5, then add cdf of lower limit with upperlimit subbed in (3)

also add F(x)= 1 is more than or equal to the upper limit

20

## mean of a rectangular distribution

### 1/2(a+b)

21

## variance

### 1/12(b-a)^2

22

## mean

### sum of x values / number of x values

23

## sample variance

###
total sum of (x values - mean)^2 / n -1

total sum of x values squared / n minus mean squared

24

## unbiased estimate of population variance

### n/n-1 x sample variance

25

## how to write confidence intervals

### (lower limit, upper limit)

26

## use Z for confidence intervals if

###
population normally distributed and population variance given

or

sample size larger than 30 even if not normally distributed as can use central limit theorem

27

## confidence interval formula

### mean + or - Z (or t) x standard deviation/ root n

28

## when to use t for confidence intervals

### population variance unknown and sample size of less than 30

29

## hypothesis testing

###
state null hypothesis (no change)

state alternate hypothesis (increase/decrease/change)

1 tailed if says increase/decrease (then take percentage to use to find Z/t value)

2 tailed if says change (then half the remianing percentage to find Z/t value)

use Z if variance known or n is more than 30

use t if variance is unknown and n is less than 30

find Z or t value (sample mean - mean / standard deviation/root n

use Z table or t table to find critical value

if Z or t value found isn't in the critical region/ more than the crictical value then accept H0 null hypothesis as not significant and no evidence to suggest the mean has changed

30

## type 1 error

### rejecting H0 and acceping H1 when H0 is actually correct

31

## type 2 error

### H0 accepted even though it is incorrect

32

## how to do a chi-squared test

###
state H0 variables independent and H1 not independent

calculate expected frequencies using matrix and chi 2way but show how to calulcate at least one of them by doing row total x column total / total

all expected frequencies must be greater than 5 and if they're not, cmobine the similar categories and recalculate the observed and expected frequencies

always convert percentages to frequencies

test stat given on calculator but show using 2 how to work it out (observed-expected)^2/expected so (O-E)^2/E

if X^2stat is greater than X^2crit then not independent

33

## How to work out X^2 stat with a 2x2 table

###
Use calculator to find out expected value

Total of (|O-E|-0.5)^2 / E

34

## area of a trapezium

### 1/2(a+b) x h

35

## P(x>2.5) and P(1.5

###
where they intersect

so find probability of P(2.5

36

## P(x>2.5 given that 1.5

### where they intersect / the second term

37