Random Variables

Question 1

Random Variable:

Answer 1

We call a function X from S to the real values R a random variable if, for every w in S, X(w) is a real number.

Question 2

Probability distribution of a random variable X

Answer 2

{P[X in B] : is a subset of R}

Question 3

Extension Theorem

Answer 3

The probability distribution of a random variable X is uniquely determined.

Question 4

Cumulative distribution function

(cdf)

Answer 4

Let X be a random variable (discrete or continuous).

Let FX(x) be a real-valued function be defined as follows:

FX(x) = P[X <= x]

for all x

Question 5

Properties of cdf

Answer 5

c) And right continuous

Question 6

Continuous random variable X

Answer 6

If FX is continuous, then the random variable X is continuous as well.

Question 7

Example cdf

Answer 7

Question 8

P[a < X <= b]

Answer 8

FX(b) - FX(b) for all a,b in R

Question 9

Discrete Random Variable

Answer 9

• cdf is a step function
• Probability mass function: pX(x) = P[X=x]
  
  • Note: pX(x) = 0 is x cannot be assumed by X
    *

Question 10

Theorem

The pmf of a discrete random variable uniquely determines the probability distribution of that random variable

Question 11

How to go from pX(x) to FX(x)?

Answer 11

Remember that pX(x) represents the “jump” height of FX(x) .

So we get pX(x0) = FX(x0) - FX(x0-)

where x0 is fixed.

Question 12

Basic properties of pX(x)

Answer 12

pX(x) >= 0

Sum over all x that X can assume pX(x) = 1

A plot of pX(x) looks like many points.

Question 13

Discrete Uniform Distribution

Answer 13

The random variable X has a discrete uniform distribution on the real numbers a1, a2, …, ak

iff

pX(ai) = P[X = ai] = 1/N for all i

Question 14

Binomial Distributions

Answer 14

The random variable X has a binomial distribution with parameters n and p

iff

P(X=x) = (nCx)(p^x)(1-p)^(n-x)

for x = 0,1,2,…,n

Question 15

Bernoulli Distribution

Answer 15

The random variable X has a Bernoulli distribution with parameter 0 < p < 1

iff

P(X=x) = p^x(1-p)^(1-x)

for x = 1,0

Question 16

Bernoulli trials

Answer 16

Only two outcomes, success and failure.

Notice: trials are independent

Question 17

What do we need to have a binomial distribution?

Answer 17

• We need to have Bernoulli trials
• The trials must be, then, independent
• The probability of a “success” has to be the same for every trial.

Question 18

Geometric Distribution

Answer 18

“Until success in the x-th trial”

The random variable X has a geometric distribution

iff

P(X=x) = (1-p)^(x-1)p

for x = 1,2,….

Question 19

Negative Binomial Distribution

Answer 19

“Trials until get x successes”

P(X=x) = (n-1)C(x-1) (1-p)^(n-1-(x-1)) p^(x-1) p

=(n-1)C(x-1) (1-p)^(n-x) p^(x-1) p

Question 20

Do example in page 36

Question 21

Posson Distribution

Answer 21

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

for x = 0,1,…

Question 22

Hypergeometric Distribution

Answer 22

Fish-in-the-lake

The random variable X has a hypergeometric distribution with parameters N, n, a

iff

P(X=x) = (aCx)*(N-aCn-x)/ (NCn)