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

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 { x1, x2, ……….. xn} or {x1, x2,……. xn}. 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
TS Inter 2nd Year Maths 2A Random Variables and Probability Distributions 1
where Pi ≥ 0 for i = 1, 2, …………. and \(\sum_{i=1}\)P1 = 1

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

→ The mean (μ) and variance (σ2) of a discrete random variable X are μ = Σxn P (X = xn) and σ2 = Σ (xn – μ)2 P ( X = xn)
= Σxn P( X – xn) – μ2
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) = nCk pk qn-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 σ2 = 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 σ2 = 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.

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