DISTRIBUCIONES DE PROBABILIDAD PARA V.A CONTINUAS:

DISTRIBUCIONES   DE PROBABILIDAD PARA V.A CONTINUAS:

A) DISTRIBUCION UNIFORME:

FDP:                

[pic 1]

FDA:

[pic 2]

Media y Varianza:

[pic 3]

Aplicaciones:                          

a) Generación de muestras de otras variables aleatorias.

b) Modela incertidumbre casi completa al ser todos valores igualmente probables en el rango [a , b] considerado.

Problemas:

1.- Let the continuous random variable X denote the current measured in a thin copper wire in mA. Assume that the range of X is [0, 20 mA], and assume that the probability density function of X is f(x) = 0.05 (0

2.-Suppose the time it takes a data collection operator to fill out an electronic form for a database is uniformly between 1.5 and 2.2 minutes.

(a) What is the mean and variance of the time it takes an operator to fill out the form?

(b) What is the probability that it will take less than two minutes

to fill out the form?

(c) Determine the cumulative distribution function of the time

3.-The thickness of a flange (attachment to another object) on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters. Determine:

(a) The cumulative distribution function of flange thickness.

(b) The proportion of flanges that exceeds 1.02 millimeters.

(c) What thickness is exceeded by 90% of the flanges?

(d) Determine the mean and variance of flange thickness.

B) DISTRIBUCION EXPONENCIAL        :

FDP:

[pic 4]

FDA:

[pic 5]

Media y Varianza:

[pic 6]

Relación con la distribución de Poisson

Si X es una variable aleatoria discreta de Poisson con media λ, que describe la tasa de ocurrencias de un evento, entonces el tiempo entre ocurrencias del evento sigue una distribución exponencial con media 1/λ.

Aplicaciones:

a) Modela tiempos de Servicio. En esa situación el parámetro λ (Media de Poisson) es una “Tasa”, de servicio es decir, atenciones / h o bien  atenciones / min… etc.

b) Modela tiempo entre llegada de entidades a un proceso de servicio, cuando  no existe relación entre llegadas sucesivas, en esa situación el parámetro λ (Media de Poisson)  es una “Tasa de llegada”, es decir, llegadas/ h o bien  llegadas / min… etc.

c) Modela Tiempos de vida de dispositivos que tienen fallas abruptas o definitivas.

Problemas:

1.-The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.

(a) What is the probability that you wait longer than one hour for a taxi?

(b) Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives within the next 10 minutes?

2.- The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own it for six years.

(a) What is the probability that the voltage regulator fails during your ownership?

(b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure?

3.-The time to failure (in hours) for a laser in a machine is modeled by an exponential distribution with λ=0.00004

(a) What is the probability that the laser will last at least 20,000 hours?

(b) What is the probability that the laser will last at most 30,000 hours?

(c) What is the probability that the laser will last between 20,000 and 30,000 hours?

4.-The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours.

(a) What is the probability that an assembly on test fails in less than 100 hours?

(b) What is the probability that an assembly operates for more than 500 hours before failure?

(c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours?

5.-The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes.

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