Teoría de Variables Aleatorias Discretas, Diapositivas de Probabilidad

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Discrete Random Variables

Random Variable

A random variable is a function

that associates a real number with

each element in the sample space

Random Variable

A random variable is called a discrete random variable if its set of possible outcomes is countable. When a random variable can take on values on a continuous scale it is called continuous random variable

Probability distribution

The set of ordered pairs (x, f(x))

is a probability function,

probability mass function, or

probability distribution of the

discrete random variable X if for

each possible outcome x,

    P  X x  f  x  f x f x x      1 0

Cumulative distribution

 (^) For any numbers a and b, where b > a 

P (a ≤ X ≤ b) = F (b) – F (a-1)

 P (X = a) = F (a) – F (a -1) 

P(X  a) = 1 – F(a-1)

Expected Value

Let X be a random variable with

probability distribution f(x). The

mean or expected value of X is

       x

 E x xf x

Variance

Let X be a random variable with

probability distribution f(x) and

mean . The variance of X is

2 2 2 2 2

     

E X E X x f x x

Standard Deviation

The positive square root of the variance, , is called the standard deviation of X

Discrete Uniform

Distribution

The mean and the variance of the

discrete uniform distribution f(x;k)

are

  k x and k x k i i k i i 2 1 2 1          

Bernoulli Trial

1. The outcome in the trial results as a success or a failure
2. The probability of success is denoted p
3. q = 1-p is the probability of failure

Binomial Distribution

The probability distribution of the

binomial random variable X, the

number of success in n trials, is

given by

  (^) p q x n x n b x n p x n x ; , ,  0 , 1 ,...,          

Binomial Distribution

The mean and the variance of the

binomial distribution b(x;n,p) are

 np and  npq

 

Hypergeometric

Distribution

The probability distribution of the Hypergeometric random variable X, the number of success in a random sample of size n selected from N items of which k are labelled success and N-k labelled failure is   x n n

N

n x N k x k h x ; N , n , k ,  0 , 1 ,..., 

Hypergeometric

Distribution

The mean and the variance of the

Hypergeometric distribution

h(x;N,n,k) are

           N k N k n N N n and N nk 1 1 2  