Random Variables and its Probability Distributions

  • A random variable is a real valued function whose domain is the sample space and Range is the set of real numbers. i.e., if X is the random variable, then the probability distribution of X is satisfies Σ P(x) = 1
  • Mean (or) Expectation of Random variable X is μ(x) = Σ x P(x)
  • Variance of random variable x is
    σ2 = Σ x2 P(x) − (Σ x P(x))2
    = Σ x2 P(x) − (Mean)2
  • Standard deviation of random variable x is
    \tt \sigma = \sqrt{variance \ of \ x}

Part1: View the Topic in this video From 03:45 To 56:18

Part2: View the Topic in this video From 01:45 To 27:37

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1. If X is a discrete random variable which assume values X1, X2, ..., Xn with respective probabilities P1, P2, ..., Pn, then the mean \overline{x} of X is defined as

                                       E(X)=\overline{X} =P_{1}X_{1}+P_{2}X_{2}+ ... + P_{n}X_{n}=\sum_{i=1}^{n} P_{i}X_{i}

2. Variance: V(X)=\sigma_x^{2}= E(X^{2})-(E(X))^{2}

     where,    E(X^{2})=\sum_{i=1}^{n} x_i^{2} P(x_{i})

3. Standard Deviation: \sqrt{V(X)}=\sigma_x= \sqrt{E(X^{2})-(E(X))^{2}}