# Likelihood Examples

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```					Likelihood Examples

Example 5.1: Suppose Yi ~ N(μ, σ 2) all i. In this case
exp{( yi  μ)2 /(2σ 2 )}
f ( yi , θ)                            i = 1,2,...,n
2πσ 2
Therefore the likelihood is the product of these
n

n                              exp{ (2σ 2 ) 1  ( yi  μ) 2 }
 f ( y , θ) 
i 1
i
i 1
(2πσ 2 ) n / 2

The loglikelihood is therefore
n                                                                                          n
n
 log( f ( y , θ))   2 log(2 )  n log( )  (2
i 1
i
2 1
)    (y
i 1
i     )2

Example 5.2: Suppose Yi ~ N (   xi , 2 ) , i = 1, 2, ..., n. The only difference from the
previous example is that the μ depend on i. Thus the loglikelihood is:

n                                                                                      n
n
 log( f ( y , θ))   2 log(2π )  n log(σ )  (2σ
i 1
i
2 1
)       (y
i 1
i    α  βxi ) 2

Bernoulli example: A Bernoulli random variable Y takes just two possible values 0 or 1:

Y = 1 with probability p
Y = 0 with probability (1 - p)

The likelihood is thus simply p if Y = 1, (1-p) if Y = 0

Thus for a sample Yi ~ Bernoulli(π (xi , β)) the likelihood is

lik     π (x , β)  (1  π (x , β))
i:Yi 1
i
j :Y j  0
j

log lik           log(π (x , β))   log(1  π (x , β))
i:Yi 1
i
j :Y j  0
j

Regression example: This is very similar to the linear regression case:

n                                                              n
n

i 1
log( f ( yi , θ))   log( 2π )  n log(σ )  (2σ 2 ) 1  ( yi  η( xi , θ)) 2
2                                   i 1
Gamma Distribution Example:
Gamma Distribution:

y α 1e  y / β
pdf: f ( y, α, β )             0 y
β α Γ(α)
cdf: No simple explicit expression

Mean = αβ
Variance = αβ 2

n
yiα 1e  yi / β
likelihood =      β α Γ(α)
i 1

n
loglikelihood =   {α log( β )  log Γ(α)  (α  1) log( y )  y / β}
i 1
i     i

n
 nα log( β )  n log Γ(α )  (α  1) {log( yi )  β 1 yi }
i 1

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