Probability - Discrete Random Variables Flashcards
what is the probability distribution of a random variable X taking K discrete values
the list of the discrete values (x) and their probabilities P(X=x)
what is the normalisation of P(X=x)
the sum from 1 to K (the number of discrete values) of the probabilities of obtaining the values is 1
what is the probability distribution closely related to
the relative frequency table
what can we do if we know P(X=x)
compute the probability of any event related to X
what is F(x) = P(X<=x)
the cumulative probability distribution, the probability that X takes a value smaller than or equal to x
how can P(X=x) and F(x) be represented
as a vertical line plot
what is E(X)
the expectation of X
what is the expectation of a random variable X
the sum from 1 to K (the number of discrete values taken by X) of the discrete values multiplied by their probabilities
which greek letter is used to represent expectation E(X)
μ
if E(X) is a weighted average of the values of X, what is it weighted according to
the probability distribution P(X=x)
what is Var(X)
the variance of X
what is Var(X)
the expected value of the random variable (X - μ)^2
what is the informal word definition of Var(X)
the weighted average of the squared distances of the values of X from μ, weighted by P(X=x)
does a large Var(X) mean a wider or narrower spread of the distribution P(X=x)
wider
what is Sd(X)
the standard deviation of X
what is the standard deviation of X
the square root of Var(X)
what 4 headings might be useful in a table being used to calculate E(X), Var(X) and Sd(X)
x, P(X=x), xP(X=x), x^2P(X=x)
what is a bernoulli trial
an action that results in one of two outcomes, (S) success or (F) failure
if P(S)=p what is P(F)=q
1-p
what is P(X=k): the probability of k successes
the binomial distribution
what is C(n, r) ‘n choose r’
the binomial coefficient
what is the binomial coefficient
the number of ways of choosing r elements from a set of n elements
what does X~B(n,p) mean
X is distributed as a binomial of n trials and success probability p
what is E(X) when X~B(n,p)
np
what is Var(X) when X~B(n,p)
np(1-p)
if X~B(n,p) when is the probability distribution of X right skewed
when p<0.5
if X~B(n,p) when is the probability distribution of X left skewed
when p>0.5
if X~B(n,p) when is the probability distribution of X symmetric
when p=0.5
if X~B(n,p) how do you find P(X>=x)
1-P(X<=x-1)
if X~B(n,p) and p>0.5 what would be a better p to work with
pfail=1-p
if X~B(n,p) how do you find P(a<=X<=b)
P(X<=b)-P(X<=a-1)
what does X~Po(λ) mean, what are X and λ
X is poisson distributed with rate λ
X gives the number of random events in a unit time/space
λ is the average rate of events per unit time/space
when would the poisson distribution be used
when the probability of success is given but the number of trials is not
if X~Po(λ) what is E(X) and Var(X)
λ
if X~Po(λ) how do you find P(a<=X<=b)
P(X<=b)-P(X<=a-1)
