S-Cool Revision Summary
S-Cool Revision Summary
A random variable is a variable which takes numerical values and whose value depends on the outcome of an experiment. It is discrete if it can only take certain values.
For a discrete random variable X with P(X = x) then probabilities always sum to 1.
P(X = x) = 1 (Remember means 'sum of').
'Cumulative' gives us a kind of running total, so a cumulative distribution function gives us a running total of probabilities within our probability table.
The cumulative distribution function, F(x) of X is defined as:
The expectation is the expected value of X, written as E(X) or sometimes as .
The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'.
To calculate the expectation we can use: E(X) = x P(X = x)
If X is a discrete random variable and f(x) is any function of x, then the expected value of f(x) is given by: E[f(x)] = f(x)P(X = x)
There are a few general results we should remember to help with our calculations of expectations: E(a) = a
E(aX + b) = aE(X) + b
The variance is a measure of how spread out the values of X would be if the experiment leading to X were repeated a number of times.
The variance of X, written as Var(X) is given by:
Var(X) = E(X2) - (E(X))2, If we write E(X) = then,
Var(X) = E(X2) - 2
Or Var(X) = E(X - )2, this tells us that Var(X) 0
There are a few general results we should remember to help with our calculations of variances:
Var(aX) = a2Var(X)
Var(aX + b) = a2Var(X)
The square root of the Variance is called the Standard Deviation of X. Standard deviation is given the symbol .
= or Var(X) = 2
Suppose that an experiment consists of n identical and independent trials, where for each trial there are 2 outcomes.
'Success' which is given probability p
'Failure' which is given probability q where q = 1 - p
Then if X = the number of successes, we say that X has a binomial distribution. We write:
Sometimes written as: X ~ bin(n, p)
If our random variable follows a binomial distribution then the associated probabilities are calculated using the following formula:
gives us the number of ways of choosing r objects from n.
It is calculated by:
You may also have a button on your calculator that will do all that for you.
If: X ~ bin(n, p)
E(X) = n x p
Var(X) = n x p xq
n = number of trials
p = probability of a success
q = probability of a failure = 1 - p
If we let X be the random variable of the number of trials up to and including the first success then X has a Geometric distribution.
The probabilities are worked out like this:
(Remember, p = probability of success, and q = probability of failure)
P(X = 1) = p
No failure, success on first attempt (hooray!)
P(X = 2) = q x p
1 failure then success
P(X = 3) = q2x p
2 failures then success
P(X = 4) = q3x p
3 failures then success
P(X = r) = qr-1x p
r - 1 failures followed by success
If X follows a Geometric distribution we write: X ~ Geo(p)
This reads as 'X has a geometric distribution with probability of success, p'.
A special result to remember: P(X > r) = qr
If: X ~ Geo(p)
E(X) = and Var(X) =