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Apuntes Estadística Introduccion a la estadística matemática, ISBN: 9789587419221
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Resumen/ formulario de la estadística dada en los primeros cursos de la carrera con estadistica.
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FormularioTema 4: Variables aleatorias
Definición de variable aleatoria
Dado un experimento aleatorio con espacio muestral Ω,
llamamos variable aleatoria a cualquier función:
𝑋: Ω → ℝ
𝐴 → 𝑋(𝐴)
A cada resultado, la variable X le asigna un número real.
Tipos de variables aleatorias
Discretas
No existen valores intermedios a los valores que toma la
variable. Ej: Las veces que vamos al baño en un día.
Continuas
Existen valores intermedios a los valores que toma la variable. Ej:
Tiempo que pasa entre cada vez que vamos al baño.
Estudio de una variable aleatoria discreta
Se definen:
1.-Posibles valores que toma la variable (𝑋𝑖 )
2.-Probabilidad con la que toman los valores anteriores (𝑝𝑖 ) →
Función de masa de probabilidad.
Función de masa de probabilidad
,Propiedades de la función de masa de probabilidad
0 ≤ 𝑃[𝑋 = 𝑥𝑖 ] ≤ 1
𝑛
∑ 𝑃[𝑋 = 𝑥𝑖 ] = 1
𝑖=1
Función de distribución de la masa de probabilidad
𝑋: ℝ → [0, 1]
𝑡 → 𝐹 (𝑡) = 𝑃[𝑋 ≤ 𝑡]
Propiedades de la función de distribución
- F es no decreciente
- F es continua por la derecha
- lim 𝐹 (𝑥 ) = 0 lim 𝐹 (𝑥 ) = 1
𝑓→−∞ 𝑓→∞
Media de una variable aleatoria discreta (Esperanza)
𝑛
𝐸 [𝑋] = ∑ 𝑥𝑖 𝑝𝑖
𝑖=1
Propiedades
𝑌 = 𝑎 + 𝑏𝑋 → 𝐸 [𝑌] = 𝑎 + 𝑏𝐸 [𝑋]
Varianza de una variable aleatoria discreta
, 𝑛 𝑛
𝑉𝐴𝑅 [𝑋] = ∑(𝑥𝑖 − 𝐸 [𝑋])2 𝑝𝑖 = ∑ 𝑥𝑖 2 𝑝𝑖 − 𝐸 [𝑋]2
𝑖=1 𝑖=1
Propiedades
𝑌 = 𝑎 + 𝑏𝑋 → 𝑉𝐴𝑅 [𝑌] = 𝑏 2 𝑉𝐴𝑅 [𝑋]
Estudio de una variable aleatoria continua
Para entenderlo mejor mostramos el ejemplo de una diana.
Probabilidad de que al disparar se le de al punto X:
𝑛º 𝑑𝑒 𝑐𝑎𝑠𝑜𝑠 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒𝑠 1
𝑃(𝑋) = = =0
𝑛º 𝑑𝑒 𝑐𝑎𝑠𝑜𝑠 𝑡𝑜𝑡𝑎𝑙𝑒𝑠 ∞
Probabilidad de que al disparar se le de a la zona amarilla:
1
𝑃(𝐴𝑚𝑎𝑟𝑖𝑙𝑙𝑎) =
4
OJO:
𝑋 = 𝛷 ⇒ 𝑃(𝑋) = 0
𝑋 = 𝛷 ⇍ 𝑃(𝑋) = 0
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