Probability - Discrete Random Variables

Question 1

what is the probability distribution of a random variable X taking K discrete values

Answer 1

the list of the discrete values (x) and their probabilities P(X=x)

Question 2

what is the normalisation of P(X=x)

Answer 2

the sum from 1 to K (the number of discrete values) of the probabilities of obtaining the values is 1

Question 3

what is the probability distribution closely related to

Answer 3

the relative frequency table

Question 4

what can we do if we know P(X=x)

Answer 4

compute the probability of any event related to X

Question 5

what is F(x) = P(X<=x)

Answer 5

the cumulative probability distribution, the probability that X takes a value smaller than or equal to x

Question 6

how can P(X=x) and F(x) be represented

Answer 6

as a vertical line plot

Question 7

what is E(X)

Answer 7

the expectation of X

Question 8

what is the expectation of a random variable X

Answer 8

the sum from 1 to K (the number of discrete values taken by X) of the discrete values multiplied by their probabilities

Question 9

which greek letter is used to represent expectation E(X)

Answer 9

μ

Question 10

if E(X) is a weighted average of the values of X, what is it weighted according to

Answer 10

the probability distribution P(X=x)

Question 11

what is Var(X)

Answer 11

the variance of X

Question 12

what is Var(X)

Answer 12

the expected value of the random variable (X - μ)^2

Question 13

what is the informal word definition of Var(X)

Answer 13

the weighted average of the squared distances of the values of X from μ, weighted by P(X=x)

Question 14

does a large Var(X) mean a wider or narrower spread of the distribution P(X=x)

Answer 14

wider

Question 15

what is Sd(X)

Answer 15

the standard deviation of X

Question 16

what is the standard deviation of X

Answer 16

the square root of Var(X)

Question 17

what 4 headings might be useful in a table being used to calculate E(X), Var(X) and Sd(X)

Answer 17

x, P(X=x), xP(X=x), x^2P(X=x)

Question 18

what is a bernoulli trial

Answer 18

an action that results in one of two outcomes, (S) success or (F) failure

Question 19

if P(S)=p what is P(F)=q

Answer 19

1-p

Question 20

what is P(X=k): the probability of k successes

Answer 20

the binomial distribution

Question 21

what is C(n, r) ‘n choose r’

Answer 21

the binomial coefficient

Question 22

what is the binomial coefficient

Answer 22

the number of ways of choosing r elements from a set of n elements

Question 23

what does X~B(n,p) mean

Answer 23

X is distributed as a binomial of n trials and success probability p

Question 24

what is E(X) when X~B(n,p)

Answer 24

np

Question 25

what is Var(X) when X~B(n,p)

Answer 25

np(1-p)

Question 26

if X~B(n,p) when is the probability distribution of X right skewed

Answer 26

when p<0.5

Question 27

if X~B(n,p) when is the probability distribution of X left skewed

Answer 27

when p>0.5

Question 28

if X~B(n,p) when is the probability distribution of X symmetric

Answer 28

when p=0.5

Question 29

if X~B(n,p) how do you find P(X>=x)

Answer 29

1-P(X<=x-1)

Question 30

if X~B(n,p) and p>0.5 what would be a better p to work with

Answer 30

pfail=1-p

Question 31

if X~B(n,p) how do you find P(a<=X<=b)

Answer 31

P(X<=b)-P(X<=a-1)

Question 32

what does X~Po(λ) mean, what are X and λ

Answer 32

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

Question 33

when would the poisson distribution be used

Answer 33

when the probability of success is given but the number of trials is not

Question 34

if X~Po(λ) what is E(X) and Var(X)

Answer 34

λ

Question 35

if X~Po(λ) how do you find P(a<=X<=b)

Answer 35

P(X<=b)-P(X<=a-1)