Revision: 12th Std >> Probability Distribution MAH-MHT CET (PCM/PCB) Probability Distribution

If the range of ( X ) is an interval (a, b), then ( X ) is called a continuous random variable.

Let ( S ) be the sample space associated with a given random experiment. Then, a real-valued function ( X ) which assigns an outcome \[w\in S\] to a unique real number X(w) is called a random variable.

If the range of the real function \[X:U\to R\]is a finite set or an infinite set of real numbers, then X is called a discrete random variable.

If a random variable X takes values x₁, x₂, …, xₙ with respective probabilities p₁, p₂, …, pₙ, then it is called the probability distribution of X.

If f(x) is the probability mass function of the random variable ( X ), then f(x) = P(X = x) is called the pmf, if

1. \[f(x)\geq0\] for all values of ( X )
2. \[\Sigma f(x)=1\]

If (X) is a continuous random variable with the pmf f(x), then its cumulative distribution function, or simply distribution function F(x), is defined as

\[F(x)=P(X\leq x)=\sum_{x_{i}\leq x}P[X=x_{i}]=\sum_{x_{i}\leq x}f(x_{i})\]

If a discrete random variable X has possible values x₁, x₂, …, xₙ with corresponding probabilities p₁, p₂, …, pₙ, then the expected value E(X) is defined as

E(X) = x₁p₁ + x₂p₂ + … + xₙpₙ = Σ xᵢpᵢ

E(X) is also called the mean, which is denoted by μ.

If (X) is a continuous random variable, then the function f(x) is called the probability density function of (X), if it satisfies

(i) \[\mathrm{f}(x)\geq0,\forall x\in\mathrm{R}\]

(ii) \[\int_{-\infty}^{\infty}\mathrm{f}(x)\mathrm{d}x=1\]

Let X be a continuous random variable with probability density function f(x). Then, the cumulative distribution function F(x) of X is defined for every real number x_i by

\[\mathrm{F}(x_{\mathrm{i}})=\mathrm{P}[\mathrm{X}\leq x_{\mathrm{i}}]=\int_{-\infty}^{x_{1}}\mathrm{f}(x)\mathrm{d}(x)\]

E(X) = Σxᵢpᵢ

Var = E(X²) − [E(X)]²

\[\mathrm{SD}(X)=\sqrt{E(X^{2})-\left[E(X)\right]^{2}}\]

SD = √Var