Probability and random variables

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PROBABILITY AND RANDOM VARIABLES

• B.TECH SEMESTER III
• 2014-18

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Continuous Random Vriable

• A continuous random variable is a random variable
  where the data can take infinitely many values. 
• For any continuous random variable with
  probability density function f(x), we will get,

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Example

• X is a continuous random variable with
  probability density function given by f(x) cx
  for 0 x 1, where c is a constant. Find c.

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System Reliability

• If T is the lifetime of a system, then
  reliability at time t is
• R(t) 1 F(t)

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Mean Time to Failure(MTTF)

• The MTFF of a system is denoted as E(T) and the
  value is

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• The failure rate of the function is denoted as
• r(t) and the value of r(t) is

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Example

• Suppose a component has a constant failure rate
  function, say r(t) ?. Find MTFF for its
  lifetime T.

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Rules for distributions

• Uniform f(x) (b is the upper
  limit and a is the lower limit)
• Exponential f(x)
• Standard normal z is a standard normal function
  and ,
• mean and standard deviation

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Joint distributions of two random variables

• f(x,y) is the joint distribution pdf,

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Marginal CDF

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Joint PDF

• Joint PDF of x and y can be denoted using f(x,y)
  and we can define it like

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Marginal PDF

• Marginal PDF of x and y are given respectively

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Checking Independence

• If any of these two cases satisfy for x and y,
  for all those cases they are independent to each
  other

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• Thank You...