Cumulative Distribution Function & Integrals Exam Practice

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In Questions 1 – 20, there is one correct answer each. Answer in the form 1 – (X1), 2 – (X2), 3 –
(X3), … (with X; € {A, B, C, D, E}) in text form or mark your choice on the exam sheet.
1. Let (t) = van lae-kods denote the cumulative distribution function of the standard normal
distribution. Which of the following statements for $(-t) is true for all t E R? [1 Point]
(A) 0(-t) = -$(t).
(B) 0(-t) = 1 – 0(t).
(C) Þ(-t) = $(t).
(D) $(-t) = -0(-t).
(E) None of the above.
2. Consider the integral
I=
cos(z)e +5+ da.
1
n
1
n
5
=
n
Which of the following expressions gives a Monte-Carlo estimate for I, which converges as n +00
in probability to I?
[1 Point]
(A) In Di= cos(x;})e-5Xi, where X1, …, Xn ~ E(1) i.i.d.
(B) In Lizz cos(X3), where X1, …, Xn ~ E(5) i.i.d.
(C) In 521-1 cos(X3), where X1, Xn ~ E(5) i.i.d.
(D) In in 1–1 cos(X?), where X1, Xn E(5) i.i.d.
(E) In 21= cos(X}), where X1, Xn ~ E(1) i.i.d.
3. Let X ~U([0, a]) with a > 0. What is the value of P[X > cx > b] for () CX > b]
(C) P[X > cx > b]
(D) P[X > c|X > b] = C-b.
(E) P[X > c|X > b] = “7.
4. Let X be a real random variable such that its law Py has probability density function
ſe(4 – 22), 2 € (0,2],
fx(x) =
x € (0,2].
a-c
a-b
=
10,
Los
What is the value of c?
[1 Point]

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Tags:
cumulative distribution

value of P

randome variable

expression and polynomials

factoring equations

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