Mathematics

user96977

3:19 PM

is the probability of tossing 3 heads in a row = 0.5^3 ?

Mew

yes

0.125

assuming the coin is unbiased

user96977

what if it is biased?

Mew

a rigged coin, a biased coin is one that is especially made so that the chance of tossing heads is greater than 50%

user96977

or less, presumably

Mew

If the coin is rigged so that the probability of heads is 80% instead of 50%, then the odds is 0.8^3

correct

user96977

3:24 PM

thanks. how does this relate to the binomial distribution?

Mew

well the binomial distribution is where you have a probability of success, p and a probability of failure, q = 1-p

you also have a certain number of trials, n

So the binomail distribution can be specified by probability of success, and number of trials

user96977

Mew

in this case, we have p = 0.5 and n = number of trials = 3

yeah

so N = number of trials = 3

user96977

thanks, i'm starting to see

Mew

P(r|f,N) means what is the probability of r successes

in thise case, n = 3 trials and we want 3 successes, so r = 3 as well

user96977

3:28 PM

so f^r is the probability of r heads, and (1-f)^(N-r) is the probability of the rest being tails

Mew

(3 3) 0.5^3 (1-0.5)^0 = 1 * 0.5^3

correct

e.g what is the probability of 2 tails occuring and 1 heads occuring:

N = 3, but this time r = 2

so P(r|f,n) = (3 2) * 0.5^2 *(1-0.5)^1

(3 2) = 3

so the answer is 3*0.5^3

in this case we still end up with 0.5^3 because (1-0.5) = 0.5

user96977

yes, i see

Mew

oops i meant 2 heads occuring and 1 tails

user96977

i do not see where the binomial coefficient comes into it though

Mew

but in this case it didn't matter

the binomial coefficient is (N r)

user96977

3:31 PM

i see where the two factors come from, and then presumably we multiply them because P(H $\cap$ T) = P(H) * P(T)

Mew

yeah

consider probability of 2 heads and 1 tails

we can get this by: HHT, HTH or THH

that is why we multiply by (3 2) = 3, because there are three ways of getting this combo

user96977

wow thanks

Mew

np

robjohn

@Chris'ssis: chat.stackexchange.com/transcript/36?m=21337791#21337791

I am going back today to move stuff back into her place. There is not too much chance that the fire will flare up now.

user96977

is there another way to use this chat, or is it only available through the browser? also, how do i enable latex for formulas in the chat

robjohn

3:41 PM

@TruthSerum I believe that a browser is the usual way to access chat. See the sidebar where it says "$\LaTeX$ in chat"

Chris's sis

@robjohn Bad news. Sorry.

user96977

@robjohn thanks, it works now

robjohn

@TruthSerum great!

user96977

4:00 PM

what is the difference between a probability density and a probability density function?

user96977

perhaps you evaluate a probability density function to obtain a probability density?

Mew

yep

what are you studying for?

user96977

just trying to broaden my knowledge. probability has always been a mystery to me. currently reading this punkuser.net/vsm/vsm_paper.pdf

user96977

"To approximate such a distribution using a small amount of data, we store the first and second moments: the mean depth and mean squared depth." - so i'm reading about moments at the moment :)

Mew

i see

E[x] is "the expectation of random variable x" which means what is the mean or average of X

simply sum all x and divide by no.

this is the 1st "moment"

user96977

4:05 PM

i see

Mew

Var[x] is the second moment

variance measures how far the values of x are from the mean

user96977

is there some generalization for the n'th moment?

Mew

yes

but it is unneccarily complicated

user96977

ha, ok

Mew

Var[x] = E[(X-u)^2]

where u is the mean , or in otherwords u = E[x]

the 3rd moment, known as skew is given by

Skew[x] = E[(X-u)^3]

user96977

4:07 PM

so Var[x] = E[(X-E[x])^2] ?

Mew

yep

so it is the expected difference between x and the mean value of x

squred

the reason it is squared is because otherwise the +ve differences would canceil with the -ve differences

so we square it to make all differences positive

the n-th moment is thus

E[(X-u)^n]

but usually only the 1st, 2nd and 3rd moments are ever really useful

mean, variance and skew

Mean = average value of X.

Skew = measure of spread of the values of X from the mean

oops i mean variance = measure of spread of the values of X from the mean

Skew: If skew is 0, the distribution is symmetric (like a bell curve). If skew is positive the distribution has a long tail to the right. If the skew is negative the distribution has a long tail to the left.

user96977

i see

user96977

in the equation for variance, what does upper-case X signify? is it the set of samples?

Mew

upper case X means "the random variable X"

So for example X could be the result of a coin flip, and thus X = 1 with 0.5 chance, and X = 0 with 0.5 chance ( if 1 = heads and 0= tails)

E[X] = expected value, or mean of the random variable X

So again if X is a coin flip this is a binomail distribution, for which E[X] = np = n*0.5.

user96977

ah, i must try to develop my idea of what this X represents

Mew

4:16 PM

X represents the outcome of a random experiment

X will follow a certain distribution. E.g. for a coin flip, X follows a binomial distribution.

user96977

or rather, what the notation E[x] means. because E is an operation defined on a discrete set of samples. as far as i understand, it is analogous to the arithmetic mean/average

Mew

yes but X is a theoretical distribution

E[X] is not the average of a sample of results

user96977

hmm, X is a random variable, and X is also a theoretical distribution?

Mew

X represents the random variable before an experiment has been concluded

have you heard of schrodinger's cat in physics?

user96977

yes

Mew

4:18 PM

well X is the state of the cat before measruement. It is both alive and dead

X is 1 with 0.5 probability and 0 with 0.5 probability

Now suppose X is the number of heads that will be achieved after 100 throws

E[X] = 100*0.5 = 50

The expected number of heads is 50, even though we haven't done an experiment

So X is the theoretical distribution of the outcome

user96977

ok, so it can be used to determine if there is a bias?

user96977

i suppose that would be one trivial use for the expected value

Mew

well, when we actually perform the experiment, we might actually count 60 heads instead of 50

So the Sample mean was 60, but the E[X] = 40

E[X] = 50 i mean

We can use E[X] and Var[X] and compare with our actual result of 40

user96977

i see. what conclusions would you make though? that might be within some kind of reasonable error for an unbiased coin

Mew

ahh

well for a coin X ~ Binomial (n, 0.5)

this means X is a random binomail distribution with n trials with success probability of 0.5

For large values of n, binomial distribution approximates the normal distribution to give:

X ~ Normal(np, npq)

This means a normal distribution with a mean of np and a variance of npq

user96977

4:23 PM

why would you need an approximation? is it cumbersome to compute with a hand calculator or something?

Mew

(X - np)/sqrt(npq) ~ N(0,1)

Now,

(60 - 50)/sqrt(100*0.5*0.5) =

2

So looking at a normal distribution table for N(0,1), the probability of getting a value of 2 or more is 0.02275

So thus there is a 2.28% chance of getting 60 heads or more with an unbiased coin

this is quite low, and thus we can conclude that the coin is probably biased

user96977

thanks, nice conclusion

Mew

yeah I wouldn't expect you to be able to do this yet, but it shows the use of E[X]

and how E[X] isn't the mean of the sample but it is the expected value given your assumptions about the distribution (in this case that it was p = 0.5)