Question

Suppose Z1, Z2, Z3, and Z4 are i.i.d. standard normal random variables. Calculate the following:

(a) P(Z1 +Z2 +Z3 +Z4 >3)

(b) P(Z12 +Z2 +Z32 +Z42 >7.78)

(c) P(Z1/?Z2 +Z32 +Z42 <1.36)

Answer 1

Answer:

Given that,

Suppose Z₁, Z₂, Z₃, and Z₄ are i.i.d. standard normal random variables.

(a).

P(Z₁ +Z₂ +Z₃ +Z₄ >3):

X =Z₁+Z₂+Z₃+Z₄ will be normal have normal distribution with

Mean =0 and

Standard deviation = =2

Probability =P(X>3)

=P(Z>(3-0)/2)

=P(Z>1.5)

=1-P(Z<1.5)

=1-0.9332

=0.0668

(b).

P(Z₁²+Z₂²+Z₃²+Z₄² >7.78):

X =Z₁²+Z₂²+Z₃²+Z₄² follow chi square distribution with 4 degree of freedom

P(X >7.78)=0.10

(c).

:

P(Z₁/√(Z₂²+Z₃²+Z₄² / 3 )) follows t distribution with 3 degree of freedom

P(X /√3 <1.36)

=P(X<2.356)

=0.95