Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 10 Random Variables and Probability Distributions to solve questions creatively.

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

**Random variable:**

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

→ Let S be a sample space and X : S → R.be a random variable. The function F : R → R defined by F(x) = P(X ≤ x), is called probability distribution function of the random variable X.

→ A set ‘A’ is said to be countable if there exists a bijection from A onto a subset of N.

→ Let S be a sample space. A random variable X : S → R is said to be discrete or discontinuous if the range of X is countable.

→ If X : S → R is a discrete random variable with range {x_{1}, x_{2}, x_{3}, ……………. } then

\(\sum_{r=1}^{\infty}\) P(x = x_{r}) = 1

→ Let X : S → R be a discrete random variable with range {x_{1}, x_{2}, x_{3}, …………….} If Σx_{r} P(X = x_{r}) exists, then Σx_{r}. P(X = x) is called the mean of the random variable X. It is denoted by µ or x . If Σ (x_{r} – µ)^{2} P(X = X_{r}) exists, then Σ(x_{r} – µ)^{2} P(X = X_{r}) is called variance of the random variable X. It is denoted by σ^{2}.

→ The positive square root of the variance is called the standard deviation of the Fandom variable x. It is denoted by σ.

**Binomial distribution:**

→ Let n be a positive integer and p be a real number-such that 0 ≤ p ≤ 1. A random variable x with range {0, 1, 2, 3, ……….. n} is said to follows (or have) binomial distribution or Bernoulli distribution with parameters n and p if P (X = r) = ^{n}C_{r} p^{r} q^{n – r} for r = 0, 1, 2, ………. n where q = 1 – p. Its Mean µ = np and variance σ^{2} = npq. n and p are called. parameters of the Binomial distribution.

**Poisson distribution:**

→ Let λ > 0 be a real, number. A, random variable x with range {0, 1, 2, ……… n} is said to follows (have) poisson distribution with parameter λ if P(X = r) = \(\frac{e^{-\lambda} \lambda^{r}}{r !}\) for r = 0, 1, 2, ……………. . Its Mean = λ and variance = λ. Its parameter is λ.

→ If X : S → R is a discrete random variable with range {x_{1} x_{2}, x_{3}, …. } then \(\sum_{r=1}^{\infty}\) P (X = x_{r}) = 1

→ Let X : S → R be a discrete random variable with range {x_{1} x_{2}, x_{3}, …..} .If Σ x_{r} P(X = x_{r}) exists, then Σ x_{r} P(X = x_{r}) is called the mean of the random variable X. It is denoted by or x.

→ If Σ(x_{r} – μ)^{2} P(X = x_{r}) exists, then Σ (x_{r} – μ)^{2} P(X = x_{r}) is called variance of the random variable X. It is denoted by σ^{2}. The positive square root of the variance is called the standard deviation of the random variable X. It is denoted by σ

→ If the range of discrete random variable X is {x_{1} x_{2}, x_{3}, …. x_{n}, ..} and P(X = x_{n}) = P_{n} for every Integer n is given then σ^{2} + μ^{2} = Σx_{n}^{2}P_{n}

**Binomial Distribution:**

A random variable X which takes values 0, 1, 2, ., n is said to follow binomial distribution if its probability distribution function is given by

P(X = r) = ^{n}c_{r}p^{r}q^{n-r}, r = 0,1,2, ……………. , n where p, q > 0 such that p + q = 1.

→ If the probability of happening of an event in one trial be p, then the probability of successive happening of that event in r trials is p^{r}.

→ Mean and variance of the binomial distribution

- The mean of this distribution is \(\sum_{i=1}^{n}\) X
_{i}p_{i}= latex]\sum_{X=1}^{n}[/latex] X.^{n}C_{x}q^{n-x}p^{X}= np, - The variance of the Binomial distribution is σ
^{2}= npq and the standard deviation is σ = \(\sqrt{(n p q)}\).

→ The Poisson Distribution : Let X be a discrete random variable which can take on the values 0, 1, 2,… such that the probability function of X is given by

f(x) = P(X = x) = \(\frac{\lambda^{x} e^{-\lambda}}{x !}\), x = 0, 1, 2, ………….

where λ is a given positive constant. This distribution is called the Poisson distribution and a random variable having this distribution is said to be Poisson distributed.