Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 10 Random Variables and Probability Distributions will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2A Random Variables and Probability Distributions Formulas

→ Bemoullian trial : If we conduct an experiment with two possible outcomes, then it is t called a Bernoullian trial.

→ Random variable : If S is the sample space of random experiment then a function X : S → R is called a random variable.

→ Discrete random variable : A random variable X whose range is either finite or countably infinite is called a discrete random variable. X is a random variable if the range of X is either { x_{1}, x_{2}, ……….. x_{n}} or {x_{1}, x_{2},……. x_{n}}. Otherwise X is said to be a continuous random variable.

→ Probability function induced by a random variable :

Suppose S is the sample space of a random experiment. Let P : P (S) → R be a probability function and X : S → R be a random variable.

Then p’ : P (R) → R defined by p’ (Y) = P(X^{-1}(Y)) for each Y ∈ P (R) is a probability function induced by X.

→ Probability distribution function : Suppose X is a random variable. Then F : R → R given by F(x) = P(X ≤ x) = P[(X^{-1}) (-∞, x) ] for each x ∈ R is called the probability distribution function of X.

→ The probability distribution of a discrete random variable is given by

where P_{i} ≥ 0 for i = 1, 2, …………. and \(\sum_{i=1}\)P_{1} = 1

→ The mean (μ) and variance (σ^{2}) of a discrete random variable X are μ = Σx_{n} P (X = x_{n}) and σ^{2} = Σ (x_{n} – μ)^{2} P ( X = x_{n})

= Σx_{n} P( X – x_{n}) – μ^{2}

The standard deviation o is the square root of the variance.

→ If p is the probability of a success, q be the probability of a failure such that p + q = 1 and n is the number of Bernoullian trials, then the probability distribution of a discrete random variable X is called a binomial variate given by P(X = k) = ^{n}C_{k} p^{k} q^{n-k}, k = 0, 1, 2, ……. n called the binomial distribution.

Here n and p are the parameters and X – B ( n, p).

→ If X – B ( n, p ) then the mean of the distribution p = np and variance of the distribution σ^{2} = npq. The standard deviation of this distribution is \(\sqrt{n p q}\).

→ The probability distribution of a discrete random variable X (called the Poisson variable) given by P(X = k) = \(\frac{\mathrm{e}^{-\lambda} \lambda^{\mathrm{k}}}{\mathrm{k} !}\) (where k = 0,1,2,…. and λ > 0) called the Poisson distribution.

Here λ Is the parameter of X.

→ If X is a Poisson variate with parameter λ then the mean μ = variance σ^{2} = λ.

→ Poisson distribution can be approximated as a limiting case of binomial distribution under the conditions.

- the number of trials must be very large, i.e., n → ∞
- ‘p’ the constant probability of success in each trial is very very small i.e., p → 0.
- n. p = λ, a finite positive real number.