概率论代考 Your answer to each question (or sub-question), should be contiguous as you will have to clearly mark where your answer is when submitting
Theory of Probability Wednesday November 11, 2020 2:00pm – 3:15pm 概率论代考
- 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, ).
- Only scientific calculators are allowed.概率论代考
- 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 概率论代考
- Binomial(n,p) distribution, k = 0, . . . , n
P[X = k] = .nΣpk(1 − p)n−k (1)
- Poisson(λ)distribution, k = 0, . . . , n
- Gamma(α,λ) distribution, x > 0
- Standardnormal distribution, −∞ < x < ∞
- Exponential(λ)distribution, x > 0
f (x) = λ e−λx (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).概率论代考
（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?概率论代考
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
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).概率论代考
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
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.
(b)[ 3 points ] ComputeE[X ].
(c)[ 4 points ] What is P[X 2 = k] for k= 0, 1, 2, . . .?概率论
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.
[ 5 points ] What is the expected value, µ, of X?
[ 5 points ] What is the variance, σ2, of X?(6)概率论代考
- [ 10 points ] Let X ∼ Binomial(123, 0.40). Use the normal approximation to the binomial distribution to estimate P[X= 42].
- [ 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?
- [ 5 points ] Let the random variable X ∼ Exponential(λ). Compute the probability density function of Y = F (X ), where F is the CDF ofX .概率论代考
- Let the random variable Z ∼ N (0,1).
- [ 7 points ] Compute the probability density function of Y =Z2.
- [ 3 points] What kind of random variable is ? (Hint: Look at the PDFs on the first page of the )
- Considerthe continuous random variable X with density function f (x) = Cxe−2x for x ≥
- [ 3 points] What is C?
- [ 3 points ] Find the distribution function F.
- [ 4 points ] What is P[X ≥1]?
- [ 10 points ] Let X ∼ Exponential(λ), and Y = log X (i.e. the natural logarithm of X ). What isthe probability density function for Y?