# Continuous Random variable

Assistant Professor at Parul University à Parul University
8 May 2015
1 sur 17

### Continuous Random variable

• 1. Continuous Random VariablesContinuous Random Variables Prepared By : Patel Jay C ME(EC)-140870705004
• 2. DefinitionsDefinitions • Distribution function: • If FX(x) is a continuous function of x, then X is a continuous random variable. o FX(x): discrete in x  Discrete rv’s o FX(x): piecewise continuous  Mixed rv’s •
• 3. DefinitionsDefinitions Equivalence: • CDF (cumulative distribution function) • PDF (probability distribution function) • Distribution function • FX(x) or FX(t) or F(t)
• 4. Probability Density Function (pdf)Probability Density Function (pdf) • X : continuous rv, then, • pdf properties: 1. 2.
• 5. DefinitionsDefinitions • Equivalence: pdf o probability density function o density function o density o f(t) = dt dF ,)( )()( 0∫ ∫ = = ∞− t t dxxf dxxftF For a non-negative random variable
• 6. Exponential DistributionExponential Distribution • Arises commonly in reliability & queuing theory. • A non-negative random variable • It exhibits memoryless (Markov) property. • Related to (the discrete) Poisson distribution o Interarrival time between two IP packets (or voice calls) o Time to failure, time to repair etc. • Mathematically (CDF and pdf, respectively):
• 7. CDF of exponentially distributedCDF of exponentially distributed random variable withrandom variable with λλ = 0.0001= 0.0001 t F(t) 12500 25000 37500 50000
• 8. Exponential Density Function (pdf)Exponential Density Function (pdf) f(t) t
• 9. Memoryless propertyMemoryless property • Assume X > t. We have observed that the component has not failed until time t. • Let Y = X - t , the remaining (residual) lifetime • The distribution of the remaining life, Y, does not depend on how long the component has been operating. Distribution of Y is identical to that of X.
• 10. Memoryless propertyMemoryless property • Assume X > t. We have observed that the component has not failed until time t. • Let Y = X - t , the remaining (residual) lifetime y t e tXP tyXtP tXtyXP tXyYPyG λ− −= > +≤< = >+≤= >≤= 1 )( )( )|( )|()(
• 11. Memoryless propertyMemoryless property • Thus Gt(y) is independent of t and is identical to the original exponential distribution of X. • The distribution of the remaining life does not depend on how long the component has been operating. • Its eventual breakdown is the result of some suddenly appearing failure, not of gradual deterioration.
• 12. Uniform Random VariableUniform Random Variable • All (pseudo) random generators generate random deviates of U(0,1) distribution; that is, if you generate a large number of random variables and plot their empirical distribution function, it will approach this distribution in the limit. • U(a,b)  pdf constant over the (a,b) interval and CDF is the ramp function
• 13. Uniform densityUniform density U(0,1) pdf 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 time cdf
• 14. Uniform distributionUniform distribution • The distribution function is given by: { 0 , x < a, F(x)= , a < x < b 1 , x > b.ab ax − −
• 15. Uniform distributionUniform distribution U(0,1) cdf 0 0.2 0.4 0.6 0.8 1 1.2 0 0.08 0.16 0.24 0.32 0.4 0.48 0.56 0.64 0.72 0.8 0.88 0.96 1 1.04 1.08 1.16 1.24 1.32 1.4 1.48 time cdf U(
• 16. a b F(x) x0 1 F(x) = 0, for x < a F(b) = 1, for x > b It is also possible to use the cumulative probability function to calculate probability. The probability is 0 for any value under a, and 1 for any value over b. F(x) = P(X ≤ x) for all x. Cumulative Distibution Function F(x)Cumulative Distibution Function F(x)
• 17. a b F(x) x To find P(x ≤ c) c c da b F(x) x To find P(c ≤ x ≤ d) P(x ≤ c) = F(c) P(c ≤ x ≤ d) = F(d) – F(c)

### Notes de l'éditeur

1. If X is to qualify as a rv, on the space (S,F,P) then we should be able to define prob. measure for X. This in turn implies that P(X &amp;lt;= x) be defined for all x. This would require the existence of the event {s| X&amp;lt;= s} belonging to F. 2. F(x) is actually absolutely continuous i.e. its derivative is well defined except possibly at the end points.
2. Sometimes we may have deal with mixed (discrete+continuous) type of rv’s as well. See Fig. 3.2 and understand it.
3. No. of failure in a given interval may follow Poisson distribution.