A discrete random variable X is said to have the Poisson distribution with parameter m > 0, if its p.m. is given by
\[P(X=x)=\frac{e^{-m}m^x}{x!}\] = 0, 1, 2, ....
A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies the following conditions:
The probability distribution of a discrete random variable X is defined by the following system of numbers.
Let the possible values of X be denoted by x1, x2, x3, . . . , and the corresponding
probabilities be denoted by p1, p2, p3, . . . , where pi = P [X = xi] for i = 1, 2, 3, . . .
Let the possible values of a discrete random variable X be denoted by x1, x2, x3, . . . ,
with the corresponding probabilities pi = P [X = xi], i = 1, 2, . . . the function p is called the probability mass function.
pi ≥ 0 and \[\sum p_i=1\]
Let X be a random variable whose possible values x1, x2, x3, . . . ,xn occur with probabilities p1, p2, p3, . . . , pn respectively. The expected value or arithmetic mean of X, denoted by E (X ) or µ, is defined by
\[\mathrm{E}(X)=\mathrm{\mu}=\sum_{i=1}^nx_ip_i=x_1p_1+x_2p_2+x_3p_3+...+x_np_n\]
In other words, the mean or expectation of a random variable X is the sum of the products of all possible values of X by their respective probabilities.
Let X be a random variable whose possible valuesx1, x2, x3, . . . ,xn occur with probabilities p1, p2, p3, . . . , pn respectively. The variance of X, denoted by Var (X ) or σ2x is defined as \[\sigma_x^2=Var\left(X\right)=\sum_{i=1}^n\left(x_i-\mu\right)^2p_i\].
A random variable is a real-valued function defined on the sample space of a random experiment.
In other words, the domain of a random variable is the sample space of a random experiment, while its co-domain is the set of real numbers.
A random variable is said to be a discrete random variable if the number of its possible values is finite or countably infinite.
The non-negative number \[\sigma_x=\sqrt{Var(X)}\] is called the standard deviation of the random variable X.
A continuous random variable differs from a discrete random variable in the sense that the possible values of a continuous random variable form an interval of real numbers.
In other words, a continuous random variable has uncountably infinite possible values.
Let Xbe a continuous random variable with the interval (a,b) as its support. The probability density function (p. d. f.) of X is an integrable function f that satisfies the following conditions:
f(x) ≥ 0 for all x∈(a,b).
For any real numbers c and d such that a ≤ c < d ≤ b,
\[P[X\in(c,d)]=\int_{c}^{d}f(x)dx\]
The probability distribution of a discrete random variable X is the set of ordered pairs
{(x1,p1), (x2,p2), (x3,p3),… }where xi are the possible values of X and pi = P(X = xi) are their corresponding probabilities, such that
pi ≥ 0 for all i,
∑pi =1
It describes all possible values of X along with their respective probabilities.
The cumulative distribution function (c. d. f.) of a continuous random variable X is defined as
\[F\left(x\right)=\int_{a}^{x}f\left(t\right)dt\]...for a < x < b.
P(X = x) = 0
A random variable is said to be a continuous random variable if the possible values of this random variable form an interval of real numbers.
A continuous random variable has uncountably infinite possible values, and these values form an interval of real numbers
The cumulative distribution function (c. d. f.) of a discrete random variable X is denoted by F and is defined as follows.
\[F\left(x\right)=P\left[X\leq x\right]\quad=\sum_{x_{i}<x}P\left[X=x_{i}\right]\]
\[=\sum_{x_i<x}P_i\]
\[=\sum_{x_i<x}f(x_i)\]
where f is the probability mass function (p. m. f.) of the discrete random variable X.
Trials of a random experiment are called Bernoulli trials if they satisfy the following
Each trial has exactly two outcomes: success or failure.
The probability of success remains the same in each trial.
The probability of x successes, P(X = x), also denoted by P(x), is given by
\[P(x)=\binom{n}{x}q^{n-x}p^x,\quad x=0,1,2,\ldots,n,\quad(q=1-p).\]
This P(x) is called the probability function of the binomial distribution.
The probability distribution of a random variable X, which represents the number of successes in n independent Bernoulli trials, each having probability of success p, is called the Binomial Distribution.
If q = 1 - p, then the probability function is given by
\[P(X=x)=\binom{n}{x}p^xq^{n-x},\quad x=0,1,2,\ldots,n.\]
Var (X ) = npq
E(X) = np
\[\sigma=\sqrt{npq}\]
