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The memory-less property is followed by which of the following continuous distribution:
1. Continuous uniform distribution
2. Normal distribution
3. Gamma distribution
4. Exponential distribution

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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 σ2 (> 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)

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