Discrete distribution

Question 1

Bernoulii random variable

Answer 1

can take two values 0 and 1

Question 2

if a random var X has Bernoulli distribution then its p.m.f. :

Answer 2

px(1) = p, px(0) = 1 - p

Question 3

Expectation and variance of Bernoulli random var

Answer 3

E(X) = p, Var(X) = p(1-p)

Question 4

Binomial distribution

Answer 4

In each case there is a fixed number of trials each have two possible outcomes called success and failures each one can be described as Bernoulli random var - we are interested in counting the total number of successes

Question 5

if a random var X has a binomial distribution with parameters n - the number of trials and p - the probability of success in each trial, then p.m.f

Answer 5

px(x) = n!/(x!(n-x)!) * p^x * (1-p)^(n-x) = (n/x)p^x*(1-p)^(n-x)

Question 6

expectation and variance of binomial distribution

Answer 6

E(X) = np
Var(X) = np(1-p)

Question 7

cumulative distribution

Answer 7

Fx(x) = P(X<=x) = sum(px(a)) from a to x

Question 8

binomial distribution in Matlab

Answer 8

p.m.f : binopdf(x, n, p)
c.d.f. : binocdf(x,n,p)
alfa quantile: binding(alfa, n, p)
binornd(nap, [1,m]) - generate m random observations from binomial distribution

Question 9

geometric distribution

Answer 9

in geometric distribution there is a sequence of success or failure Bernoulli trials, but the number of trials is not fixed

Question 10

p.m.f of geometric distribution

Answer 10

px(x) = (1-p)^(x-1)*p

Question 11

expectation and variance

Answer 11

E(X) = 1/p
Var(X) = (1-p)/p^2

Question 12

poisson distribution

Answer 12

used to represent more general count data; the number of times an event occurs in a finite interval in time or space - used mainly when events is regarded as rare

Question 13

p.m.f of Poisson distribution with rate parameter b

Answer 13

px(x) = P(X=x) = (e^-b*b^x)/x!

Question 14

expectation and variance of Poisson distribution

Answer 14

E(X) = VAR(X) = b

Question 15

the Poisson distribution in Matlab

Answer 15

pm.f - poisspdf(x, lambda)
c.d.f - poisscdf(x, lambda)
alfa quantile - poising(alfa, lambda)

Question 16

joint probability mass function of two discrete random variables X and Y

Answer 16

p.m.f(x,y) = P{(X=x) & (Y=y)}

Question 17

marginal distribution of X’s p.m.f. and Y’s p.m.f.

Answer 17

px(x) = sum(px,y(x,y))
py(y) = sum(px,y(x,y))

Question 18

multinomial distribution

Answer 18

this is the generalisation of binomial distribution to the case where there are more than two possible outcomes in each trial - we have n trials and k - possible outcomes

Question 19

p.m.f of multinomial distribution with n trials and probabilities p1, p2, …, pn

Answer 19

px(x1, x2, x3, … , xn) = (n!/x1!x2!…xn!)p1^x1p2^x2…*pn^xn

Question 20

conditional distribution for multinomial random variable

Answer 20

(X2,….,Xn)|X1 = multinomial(n-X1; p2/(1-p1),…,pn/(1-p1))

Question 21

what does it mean that variables are identically distributed

Answer 21

if n random variable have the same probability distribution

Question 22

Expectation and variance of independent and identically distributed random variables

Answer 22

E(X’(n)) = a
Var(X’(n)) = b^2/n
where X’(n) to be the mean:
X’(n) = sum(Xi/n)

Question 23

standard error

Answer 23

sqrt(Var(X’(n)) = b/sqrt(n),

denoted by SE(X’(n))

Question 24

central limit theorem

Answer 24

for random variables X1, X2…,Xn each with mean a and variance b^2, the distribution of X’(n) tends to normal distribution with mean a and var b^2 as n-> inf:
X’(n) = N(a, b^2/n) or
sum(Xi) = N(na, nb^2)