Probability theory and mathematical statistics-ch2-random variables and their distributions

1. Random variables

We do not care about the specific results or the order of occurrence of random trials, and the relationship is the number associated with them.

Random variables are a function of linking the result of a random experiment with a number, and its value has a certain probability.

The random variables are in uppercase letters, and the real numbers are in lowercase letters.

Such as: Throw three times coins appear 2 times the probability of positive.

The result of random test A is ={positive and negative, just anyway, just}

Random variable x=2

Probability =3/8

Then P (x=2) =p (A) is P (positive and negative, just anyway, anyway positive) =3/8

2. Discrete random variables and their distribution laws

Some random variables, all of which may take different values are finite or can be listed infinitely, this random variable is called a discrete random variable.

Distribution law of discrete random variable x:

Set X as a discrete random variable, all of which may be valued as XK (k=1,2,....), X takes the probability of each possible value,
p{x=xk}=pk,k=1,2 ...
The distribution law of the X is called the formula.

2 conditions are met:

pk≥0;

∑pk=1.

Important Distribution:

Two items distributed:

The test e has only two possible results: A and a, then E is the Bernoulli test. P (a) =p,p (a) =1-p

N Heavy Bernoulli test: The test E is repeated n times independently.

Independence: The results of each test do not affect each other; repetition: the probability that event a occurs does not change.

X indicates the number of occurrences of event A in the N-Bernoulli test.

When N=1, that is, only 1 Bernoulli test, random variable x can only take 0,1 value.

It is called a two-item distribution because the upper formula is an item of the two-item (P+Q) n.

Poisson distribution:

3. Distribution function of random variables

For continuous random variables, it doesn't make sense to see the probability value at a certain point (in fact, the probability at any point is =0). So we should study the probability that the value of continuous random variable falls into an interval.

Distribution functions:

Basic properties:

F (x) is a non-decreasing function;

The value is between 0 and 1;

F (x+0) =f (x), i.e. f (x) is right continuous.

F (x) is the cumulative probability value of the x≤x.

The distribution function of the discrete random variable x:

4. Continuous type random variable and its probability density

The probability of a continuous type of random variable being evaluated at any point is 0. As a corollary, the probability of a continuous random variable taking value on the interval is independent of whether the interval is open or closed interval. Note that the probability is p{x=a}=0, but {x=A} is not an impossible event.

Important Distribution:

Evenly distributed

Exponential distribution:

Or

An important feature of exponential functions is the memory-free nature. This means that if a random variable is exponentially distributed, there is P (t>s+t| when s,t≥0) t>t) =p (t>s) Normal distribution:

Properties:

Normal curve about X=µ symmetry;

When the X=µ is taken to the maximum value, the normal curve begins at the place where the mean value is, and gradually drops evenly to the left and right sides respectively.

At the Μ±σ Place has the inflection point;

Its probability density and distribution function are recorded as: φ (x), φ (x)

Lemma:

Standardized transformations

Normal distribution, as soon as possible the range of normal variables is (-∞,∞), but the probability of falling into (µ-δ,µ+δ) is 68.26% (that is: f (x) in the area of this interval), the probability of falling into (µ-3δ,µ+3δ) is 99.74%, almost affirmative, this is the 3δ law.

Upper ª: For a standard normal distribution , if zª satisfies the condition P (x>zª) =ª, then zª is the upper ª sub-site of the standard overall distribution.

5. Distribution of functions of random variables

Discusses how to derive the probability distribution of its function y=g (x) from the probability distribution of a known random variable x.

Probability theory and mathematical statistics-ch2-random variables and their distributions