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

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 {x1, x2, x3, ……………. } then
$$\sum_{r=1}^{\infty}$$ P(x = xr) = 1

→ Let X : S → R be a discrete random variable with range {x1, x2, x3, …………….} If Σxr P(X = xr) exists, then Σxr. P(X = x) is called the mean of the random variable X. It is denoted by µ or x . If Σ (xr – µ)2 P(X = Xr) exists, then Σ(xr – µ)2 P(X = Xr) is called variance of the random variable X. It is denoted by σ2.

→ 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) = nCr pr qn – r for r = 0, 1, 2, ………. n where q = 1 – p. Its Mean µ = np and variance σ2 = 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 {x1 x2, x3, …. } then $$\sum_{r=1}^{\infty}$$ P (X = xr) = 1

→ Let X : S → R be a discrete random variable with range {x1 x2, x3, …..} .If Σ xr P(X = xr) exists, then Σ xr P(X = xr) is called the mean of the random variable X. It is denoted by or x.

→ If Σ(xr – μ)2 P(X = xr) exists, then Σ (xr – μ)2 P(X = xr) is called variance of the random variable X. It is denoted by σ2. 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 {x1 x2, x3, …. xn, ..} and P(X = xn) = Pn for every Integer n is given then σ2 + μ2 = Σxn2Pn

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) = ncrprqn-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 pr.

→ Mean and variance of the binomial distribution

• The mean of this distribution is $$\sum_{i=1}^{n}$$ Xipi = latex]\sum_{X=1}^{n}[/latex] X. nCxqn-xpX = np,
• The variance of the Binomial distribution is σ2 = 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.
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