Let P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find it
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Let P be the standard normal distribution, i.e., P is the proba-
bility measure on R, B(R) given by
dP(x) = 1
√2π
e
− x2/2
dx.
Consider the random variables
fn(x) = (1 + x2) 1/n
e^(x^2/n+2)
x ∈ R, n ∈ N.
Using the dominated convergence theorem, prove that the limit
limn→∞
E(fn)
exists and find it
Step by step
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