Transcript Section 7 – Continuous Distributions

Section 7 – Continuous
Distributions
Uniform
1
: for(a  x  b)
(b  a)
 0 : x  a
x  a
F(x)  
:a x b
b  a
 1: x  b
ab
E[X] 
2
(b  a) 2
Var[X] 
12
f (x) 
• The probability of each
X in the interval is
“uniform” (the same)
Uniform can be discrete (ex: dice only have

integers) or continuous
(all values in interval)
Uniform: Discrete vs. Continous
Discrete
1
p(x) 
N


N 1
E[X] 
2
N 2 1
Var[X] 
12
Continuous
1
f (x) 
: for(a  x  b)
(b  a)
 0 : x  a
x  a
F(x)  
:a x b
b  a
 1: x  b
ab
E[X] 
2
(b  a) 2
Var[X] 
12
Normal Distribution
• Notation: X ~ N(mean, variance)
– Note: the second number is variance not standard
deviation
– Ex: Standard Normal, Z ~ N(0, 1)
• You will likely not need to know the pdf
– You will be given a normal table to find common z
values
• One-sided: use the (1 – alpha) percentile
• Two-sided: use the (1 – alpha/2) percentile
– P(r<X<s) = P[(r-µ)/σ < (X-µ)/σ < (s-µ)/σ ]
• Be comfortable using the table!!!
Normal Approximation of Other
Distribtutions
• Given RV X, mean, and variance of the
distribution (without knowing what the real
distribution is)
• Use normal distribution with the same
mean/variance to approximate the true
probability
• Integer Correction for Discrete Distributions
– P(n<=X<=m) becomes P(n-1/2<=X<=m+1/2)
– Explained/justified well in Actex (see p. 200)
– You’ll likely still be closest to the right answer without
this, but it’s more accurate this way
When do I divide by sqrt(n)?
• Estimating a value (X)
X ~ N(X , X2 )
– No square root of n
• Estimating a mean (X bar) given a
sample size

– Involves square root of n
– (ex: SOA 123 #81)
– Hint: you only use n when given a
sample size, and it’s used to
decrease the size of the interval b/c
an average is less variable
• Note: the sample size does not
affect the mean, only the variance
 mean 
X
n
2
X ~ N(x , mean
)
X ~ N(x ,

2
X
n
)
Exponential Distribution
• Usage: X is time until
an event occurs
• Parameter: Lamba
(mean = 1/Lambda)
– Alternative: Use
Theta = 1/Lambda
• Theta = Mean
• Can rewrite pdf, E[X],
Var[X], etc. using
Theta

f (x)  e x for : x  0
F(x)  1 e x for : x  0
E[X] 
1

Var[X] 
1
2
“Memoryless” Property
• Concept: what happened before
doesn’t affect what’s going to
happen now
• Exponential is about time until
an event occurs
• Ex: if no insurance claims
have happened in the past
month, the exponential
doesn’t think that one is
“due” now
• There is just as much chance
of a claim happening this
week as there was in the
week following the first
claim
P[X  x  y | X  x]
P[X  x  y  X  x]

P[X  x]
P[X  x  y]

P[X  x]
e (x y )
  x
e
 e  y
 P[X  y]
Link Between Exponential and Poisson
Distributions
• These distributions are connected by the same
parameter: Lambda
• X is the time between events (continuous)
– time per events
– X ~ Exponential, with mean (1/Lambda)
– Ex: time between claims
• N is the number of events that have occurred
while that time elapsed (discrete)
– events per time
– N ~ Poisson, with mean Lambda
– Ex: number of claims in a period of time
Minimum of Multiple Exponential RV’s
• Given multiple RV’s with exponential distributions and their means
(1/Lambda), find some probability involving the minimum of all of
the RV’s (the lowest value of all of the RV’s  the time at which the
first event occurs)
– An RV with a higher mean may still occur before the lower means (due
to randomness)
• Trap: do not just add the means of the exponential distributions
• Technique
– Convert to Poisson distributions, each with mean Lambda
– Add the Lambda’s
• This new Lambda is the parameter for Y = min{Y1, Y2, …, Yn}
• Y ~ exponential with mean (1/Lambda)
• Key Point: don’t add the means of the exponentials, convert to
lambda’s, add the lambda’s, convert back to exponential
• This is a harder problem, but VERY COMMONLY TESTED
Gamma Distribution
f(x) = βα*xα-1*e-βx
Γ(α)
Γ(n) = (n-1)! (if n is a positive integer)
E(X) = α/β
Var(X) = α/β2
• Actex does not recognize this as an important distribution to know
• Uniform, Normal, and Exponential are the MOST important