Fall 2020 MATH 233

Exam 2

Theory of Probability

Wednesday November 11, 2020 2:00pm – 3:15pm

概率论Exam代写 Nationwide, the Math SAT exam has an average score of 500 (out of 800) and a standard deviationof 120 (assuming a normal distribution).

Your answer to each question (or sub-question), should be contiguous as you will have to clearly mark where your answer is when submitting to Gradescope.

• No collaboration allowed.
• Nooutside sources allowed (no notes, no books, etc.).
• Only scientific calculators are allowed.
• Tablesof the standard normal PDF and CDF are included.
• Nocredit will be given without showing ALL work.
• Pleaseplace a box around your answer for each question.

A scanned PDF must be uploaded to Gradescope by 3:25pm. No late submissions will be accepted.

Useful Random Variables  概率论Exam代写

• Binomial(n,p) distribution, k = 0, . . . , n

(1)

• Poisson(λ)distribution, k = 0, . . . , n

(2)

• Gamma(α,λ) distribution, x > 0

(3)

• Standardnormal distribution, −∞ < x < ∞

(4) 

• Exponential(λ)distribution, x > 0

 (5)

1.Nationwide, the Math SAT exam has an average score of 500 (out of 800) and a standard deviationof 120 (assuming a normal distribution).

概率论Exam代写

(a)[ 5 points ] If five students who took the exam are selected at random, what is the probabilitythat exactly two of them scored at least 680?

(b)[ 5 points ] What is the probability that a randomly chosen senior scored less than400?

2.[ 5 points ] An insurance company insures 3000 people, each of whom will be in an accident with probability 0.001. Approximate the probability that the insurance company will have to payout on at most 2 accidents.  概率论Exam代写

3.[ 5 points ] The median of a random variable X is the value x such that P[X ≤ x] = 0. Compute the median of X ∼Exponential(3).

4.[ 10 points ] The mode of a continuous random variable is the value x which maximizes f (x), where f is the probability density function of . Find the mode of a Gamma(4.5, 0.5) random variable.

5.Considerthe discrete random variable X with probability mass function P[X = n] = C 3−|n|, for n = . . . , −2, −1, 0, 1, 2, . . ..

(a)[ 3 points ] Find C.  概率论Exam代写

(b)[ 3 points ] ComputeE[X ].

(c)[ 4 points ] What is P[X 2 = k] for k= 0, 1, 2, . . .?

6. Considerthe discrete random variable X defined by its probability mass function:

P[X = 0] = 0.20,        P[X = 2] = 0.10,

P[X = 1] = 0.20,         P[X = 3] = 0.50.  (6)

(a)[ 5 points ] What is the expected value, µ, of X?

(b)[ 5 points ] What is the variance, σ², of X?

7.[ 10 points ] Let X ∼ Binomial(123, 0.40). Use the normal approximation to the binomial distribution to estimate P[X= 42].   概率论Exam代写

8.[ 5 points ] Let Z be a standard normal random variable, with PDF denoted by ϕ and CDF denoted by Φ. Let X = Φ(Z). How is Xdistributed?

9.[ 5 points ] Let the random variable X ∼ Exponential(λ). Compute the probability density function of Y = F (X ), where F is the CDF ofX .

10.Let the random variable Z ∼ N (0,1).

(a)[ 7 points ] Compute the probability density function of Y =Z2.

(b)[  3  points] What kind of random variable is ? (Hint: Look at the PDFs on the first page of the exam. )

11.Considerthe continuous random variable X with density function f (x) = Cxe^−2x for x ≥ 0. 概率论Exam代写

(a)[ 3 points] What is C?

(b)[ 3 points ] Find the distribution function F.

(c)[ 4 points ] What is P[X ≥1]?

12.[ 10 points ] Let X ∼ Exponential(λ), and Y = log X (i.e. the natural logarithm of X ). What isthe probability density function for Y?

Standard normal PDF

Standard normal CDF