STAT 100A HWIV Due next Wed in class
Problem 1: For Z Bernoulli(p), calculate E[Z ]. Problem 2: For X Binomial(n, p), calculate E[X ]. Problem 3: For X Geometric(p), calculate E[X ]. Problem 4: Suppos
Statistics 100A
Homework 5 Solutions
Ryan Rosario
Problem 1 For a continuous variable X f (x). For each part, it is not sucient to simply use
the properties of expectation and variance.
(1) Prove E [a
STAT 100A HWI
Note (1) If all the outcomes are equally likely, then Pr(A) = |A|/|. (2) Conditional probability Pr(A|B ) = Pr(A B )/ Pr(B ). Please show all the necessary steps in your calculations. Pl
STAT 100A HWII
Note
(1) Rule of total probability. Suppose A1 , A2 , ., An partition the sample space, then P (B ) =
n
n
i=1 P (Ai B ) =
i=1 P (Ai )P (B |Ai ).
(2) Bayes rule. P (Ai |B ) = P (Ai B )/P
STAT 100A HWIII
Problem 1: Suppose 1% of the population is inicted with a particular disease. For a medical test,
if a person has the disease, then 95% chance the person will be tested positive. If a
STAT 100A HWIV
Problem 1: For a discrete random variables X ,
(1) Prove E[aX + b] = aE[X ] + b.
(2) Prove Var[aX + b] = a2 Var[X ].
(3) Let = E[X ] and 2 = Var[X ]. Let Z = (X )/ , calculate E[Z ] and
STAT 100A HWI Due next Wed in class
Problem 1: Suppose we ip a fair coin 4 times independently. (1) What is the sample space? (2) What is the set that corresponds to the event that the number of heads
STAT 100A HWI Solution
Problem 1: Suppose we ip a fair coin 4 times independently. (1) What is the sample space? A: The sample space consists of all the 24 = 16 sequences of heads and tails. (2) What
STAT 100A HWII Due next Wed in class
Problem 1: If we flip a fair coin n times independently, what is the probability that we observe k heads? k = 0, 1, ., n. Please explain your answer. Problem 2: Pr
STAT 100 Homework II Answer Key
Problem 1: Let X and Y be the two numbers respectively. p(X > 3) = 1/3. p(X > 3|X + Y > 9) = 1. p(X = k|X Y | > 2) = p(X = k, |X Y | > 2)/p(|X Y | > 2). p(|X Y | > 2) =
STAT 100A HWII Solution
Problem 1: If we ip a fair coin n times independently, what is the probability that we observe k heads? k = 0, 1, ., n. Please explain your answer. A: The probability is n /2n
STAT 100A HWIII Due next Wed in class
Problem 1: Suppose an urn has r red balls and b blue balls. We random pick a ball, and then we put three balls of the same color back to the urn. After that we ra
Statistics 100A
Homework 4 Solutions
Ryan Rosario
Problem 1
For a discrete random variable X,
Note that all of the problems below as you to prove the statement. We are proving the properties
of expect
STAT 100A HWIII Solution
Problem 1: Suppose 1% of the population is inicted with a particular disease. For a medical test,
if a person has the disease, then 95% chance the person will be tested positi
STAT 100A HWIV Solution
Problem 1: For Z Bernoulli(p), calculate E[Z]. A: E[Z] = 0 (1 p) + 1 p = p. Problem 2: For X Binomial(n, p), calculate E[X]. A: We can represent X = Z1 + Z2 + . + Zn , where Zi
STAT 100A HWV Due next Wed in class
Problem 1: For a discrete random variable X , prove (1) E[aX + b] = aE[X ] + b. (2) Var[aX + b] = a2 Var[X ]. Problem 2: For two discrete random variables X and Y ,
STAT 100A HWV Solution
Problem 1: For a discrete random variable X, prove (1) E[aX + b] = aE[X] + b. A: Let p(x) be the probability mass function of X. Then E[aX + b] = x (ax + b)p(x) = a x xp(x) + b
STAT 100A HWVI Due next Wed
Problem 1: Suppose we flip a fair coin n times independently. Let X be the number of heads. Let k = n/2 + z n/2, or z = (k - n/2)/( n/2). Let g(z) = P (X = k). 2 (1) Using
STAT 100A HWVI Solution
Problem 1: Suppose we ip a fair coin n times independently. Let X be the number of heads. Let k = n/2 + z n/2, or z = (k n/2)/( n/2). Let g(z) = P (X = k). 2 (1) Using the Stir
STAT 100A HWVII Due next Fri
Problem 1: Suppose Z N(0, 1). The density of z is f (z) = 1 e-z /2 . E[Z] = 0, Var[Z] = 1. 2 Let X = + Z, where > 0. (1) Find the probability density function of X. (2) Ca
STAT 100A HWVII Solution
Problem 1: Suppose Z N(0, 1). The density of z is f (z) = Let X = + Z, where > 0. (1) Find the probability density function of X. A: Let g(x) be the density of X, then g(x) =
STAT 100A MIDTERM EXAM Solution
Problem 1:Suppose we generate two independent random variables X and Y uniformly over [0, 1]. (1) (4 points) Calculate P (X 2 + Y 2 1). A: Let be the unit square [0, 1]
STAT 100A Review for midterm
Note: The following are the materials to be covered in the midterm.
1
Basic concepts
When an experiment is performed, the outcome can be random. The sample space is the se
STAT 100A Review for final
1 Part I: study one random variable at a time
A discrete random variable X takes values in a discrete list cfw_x1 , x2 , .. Its behavior is governed by a probability mass fu
Statistics 100A
Homework 1 Solutions
Ryan Rosario
Problem 1
Suppose we ip a fair coin 4 times independently.
(1) What is the sample space?
By denition, the sample space, denoted as , is the set of all
Statistics 100A
Homework 2 Solutions
Ryan Rosario
Problem 1
Suppose an urn has b blue balls and r red balls. We randomly pick a ball. If the ball is red, we put
two red balls back to the urn. If the b
STAT 100A HWIII Solution
Problem 1: Suppose an urn has r red balls and b blue balls. We random pick a ball, and then we put three balls of the same color back to the urn. After that we randomly pick a