{"id":22081,"date":"2024-01-10T16:04:16","date_gmt":"2024-01-10T10:34:16","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22081"},"modified":"2024-01-11T12:20:36","modified_gmt":"2024-01-11T06:50:36","slug":"inter-2nd-year-maths-2a-random-variables-and-probability-distributions-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-random-variables-and-probability-distributions-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A Random Variables and Probability Distributions Formulas"},"content":{"rendered":"

Use these Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 10 Random Variables and Probability Distributions to solve questions creatively.<\/p>\n

Intermediate 2nd Year Maths 2A Random Variables and Probability Distributions Formulas<\/h2>\n

Random variable:<\/strong>
\n\u2192 Suppose S is the sample space of a random experiment then any function X : S \u2192 R is called a random variable.<\/p>\n

\u2192 Let S be a sample space and X : S \u2192 R.be a random variable. The function F : R \u2192 R defined by F(x) = P(X \u2264 x), is called probability distribution function of the random variable X.<\/p>\n

\u2192 A set ‘A’ is said to be countable if there exists a bijection from A onto a subset of N.<\/p>\n

\u2192 Let S be a sample space. A random variable X : S \u2192 R is said to be discrete or discontinuous if the range of X is countable.<\/p>\n

\"Inter<\/p>\n

\u2192 If X : S \u2192 R is a discrete random variable with range {x1<\/sub>, x2<\/sub>, x3<\/sub>, ……………. } then
\n\\(\\sum_{r=1}^{\\infty}\\) P(x = xr<\/sub>) = 1<\/p>\n

\u2192 Let X : S \u2192 R be a discrete random variable with range {x1<\/sub>, x2<\/sub>, x3<\/sub>, …………….} If \u03a3xr<\/sub> P(X = xr<\/sub>) exists, then \u03a3xr<\/sub>. P(X = x) is called the mean of the random variable X. It is denoted by \u00b5 or x . If \u03a3 (xr<\/sub> – \u00b5)2<\/sup> P(X = Xr<\/sub>) exists, then \u03a3(xr<\/sub> – \u00b5)2<\/sup> P(X = Xr<\/sub>) is called variance of the random variable X. It is denoted by \u03c32<\/sup>.<\/p>\n

\u2192 The positive square root of the variance is called the standard deviation of the Fandom variable x. It is denoted by \u03c3.<\/p>\n

Binomial distribution:<\/strong>
\n\u2192 Let n be a positive integer and p be a real number-such that 0 \u2264 p \u2264 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<\/sup>Cr<\/sub> pr<\/sup> qn – r<\/sup> for r = 0, 1, 2, ………. n where q = 1 – p. Its Mean \u00b5 = np and variance \u03c32<\/sup> = npq. n and p are called. parameters of the Binomial distribution.<\/p>\n

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Poisson distribution:<\/strong>
\n\u2192 Let \u03bb > 0 be a real, number. A, random variable x with range {0, 1, 2, ……… n} is said to follows (have) poisson distribution with parameter \u03bb if P(X = r) = \\(\\frac{e^{-\\lambda} \\lambda^{r}}{r !}\\) for r = 0, 1, 2, ……………. . Its Mean = \u03bb and variance = \u03bb. Its parameter is \u03bb.<\/p>\n

\u2192 If X : S \u2192 R is a discrete random variable with range {x1<\/sub> x2<\/sub>, x3<\/sub>, …. } then \\(\\sum_{r=1}^{\\infty}\\) P (X = xr<\/sub>) = 1<\/p>\n

\u2192 Let X : S \u2192 R be a discrete random variable with range {x1<\/sub> x2<\/sub>, x3<\/sub>, …..} .If \u03a3 xr<\/sub> P(X = xr<\/sub>) exists, then \u03a3 xr<\/sub> P(X = xr<\/sub>) is called the mean of the random variable X. It is denoted by or x.<\/p>\n

\u2192 If \u03a3(xr<\/sub> – \u03bc)2<\/sup> P(X = xr<\/sub>) exists, then \u03a3 (xr<\/sub> – \u03bc)2<\/sup> P(X = xr<\/sub>) is called variance of the random variable X. It is denoted by \u03c32<\/sup>. The positive square root of the variance is called the standard deviation of the random variable X. It is denoted by \u03c3<\/p>\n

\u2192 If the range of discrete random variable X is {x1<\/sub> x2<\/sub>, x3<\/sub>, …. xn<\/sub>, ..} and P(X = xn<\/sub>) = Pn<\/sub> for every Integer n is given then \u03c32<\/sup> + \u03bc2<\/sup> = \u03a3xn<\/sub>2<\/sup>Pn<\/sub><\/p>\n

Binomial Distribution:<\/strong>
\nA random variable X which takes values 0, 1, 2, ., n is said to follow binomial distribution if its probability distribution function is given by
\nP(X = r) = n<\/sup>cr<\/sub>pr<\/sup>qn-r<\/sup>, r = 0,1,2, ……………. , n where p, q > 0 such that p + q = 1.<\/p>\n

\u2192 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<\/sup>.<\/p>\n

\u2192 Mean and variance of the binomial distribution<\/p>\n