Correct Answer - Option 4 : Exponential distribution

**Explanation:**

Memoryless property = A variable X is memoryless with respect to t if for all s with t ≠ 0

⇒ P(x > s + t I x > t) = P(x > s)

⇒ P(x > t, x > T)/P(x > T) = P(x > s)

⇒ P(x > s + t) = P(x > s) P(x > t)

This property is satisfied by exponential distribution

P(x > t) = e^{-λt}

⇒ P(x > s + t) = e^{-λ(s + t)}

∴ P(x > s + t) = P(x > s) P (x > t)

⇒ e^{-λs} × e^{-st}

⇒ e^{-λ(s + t)}

**∴ This is only memoryless random distribution **

The only continuous probability distributions that are memoryless are the exponential distributions. If a continuous X has the memoryless property

(over the set of reals) X is necessarily an exponential.

**Continuous uniform distribution** = A random variable X is said to follow a continuous uniform (rectangular) distribution over an interval (a, b) if its probability density function is given by

\(f(x) = \left\{ \begin{matrix} \dfrac{1}{b-a} \ \text{for} \ a < x < b \\\ 0 , \ \ \text{otherwise} \end{matrix} \right.\)

**Normal distribution** = A continuous random variable X is said to follow normal distribution with parameters **μ(- ** ∞ < μ < **∞) **and σ^{2} (> 0)** **if it takes on any real value and its probability density function is given by

\(f(x)=\dfrac{1}{\sqrt[\sigma]{2\pi}} e^{\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\)

**Gamma distribution = **if n > 0, the integral \(\displaystyle\int_0^\infty\) is called gamma function and t is denoted by 1(n̅)