21. Discrete random variable

Discrete random variable - data that’s set of possible outcomes are countable or can be represented as whole numbers.

Def. 1 A random variable is a function, which associate a real number with each element in the sample space.

Def 2. The set of ordered pairs (x, f(x)) is a probability function or a probability mass function or a probability distribution of a discrete random variable X for each possible outcomes is equal:

f(x)≥0

∑f(x)=1

P(X=x)=f(x)

Probability function has some properties:

1. 0≤f(x)≤1

2. f(x1<x<x2)=f(x2)-f(x1)

3. Probability distribution is non-decreasing f(x1)=f(x2), for x2<x1

4. It is continuous on the right

lim f(x)>0

lim f(x)=1

Def 3. Cumulative distribution of function or briefly the distribution function for a random variable X is defined by F(x)= P(X≤x). Where x-real number

-∞<x<∞ the distribution

F(x) has the following properties:

1. F(x) non-decreasing: F(x1)≤F(x2), for x1≤x2

2.

3. F(x) is continuous from the right Ɏh=R

(x→+0) lim F(x+h)=F(x)

Def 4. Distribution function for discrete random variable. The distribution function is obtain from the probability function f(x) by noting for all x: -∞<x<∞

F(x)= P(X≤x)=∑ f(x)

Where the sum is taking over for all values u on by X for which

u€X u≤x

if the X taken only a finite number of values x₁, x₂, x₃…x_n then distribution function

0 -∞<x<x₁

F(x)= f(x₁) x₁≤x<x₂

f(x₁)+f(x₂) x₂≤x<x₃

f(x₁)+ f(x₂)+ f(x₃)+ …+f(x_n) x_n≤x<+∞

Example: suppose that a coin is tossed twice so that the sample space S

S= {HH,HT,TH,TT}

Sample space	HH	HT	TT
x	2	1	0

P(HH)= ¼

P(HT)= ½

P(TT)= ¼

x	2	1 1	00 0
F(x)	¼ ¼	½	¼

0 -∞<x<0

F(x)= ¼ 0≤x<1

¾ 1≤x<2

1 2≤x<+∞

Distribution function

22. Discrete Probability Distributions. Probability Density function.

A discrete probability distribution shall be understood as a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is then called a discrete random variable, if

as u runs through the set of all possible values of X. It follows that such a random variable can assume only a finite or countably infinite number of values.

Def: Let X be a random variable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. A discrete probability distribution consists of the values of the random variable X and their corresponding probabilities P(X).  The probabilities P(X) are such that 

∑ P(X) = 1

Probability density function 

In probability theory, a probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

A random variable X with values in a measure space   (usually Rⁿ with the Borel sets as measurable subsets) has as probability distribution the measure X_∗P on  : the density of X with respect to a reference measure μ on   is the Radon–Nikodym derivative:

That is, f is any measurable function with the property that:

for any measurable set  .