## Probability

# 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

σ^{2}= Σ x^{2}P(x) − (Σ x P(x))^{2}

= Σ x^{2}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 X_{1}, X_{2}, ..., X_{n} with respective probabilities P_{1}, P_{2}, ..., P_{n}, 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}}