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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

comment on one group ... in chi squared

mention observed compared to expected values