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 function p(x) for x = cfw_x1 , x2 , ., so that P (X = x)
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 set of all the possible outcomes. It is often denoted by
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]2 , and let A be the event that X 2 + Y 2 1, then P (A)
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) = =
2 1 ez /2 . 2
E[Z] = 0, Var[Z] = 1.
P (Z (z, z + z) P
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) Calculate E[X] and Var[X]. Problem 2: Suppose U Uniform(0
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 Stirling formula n! 2nnn en , show that g(0) 1 n . a b mean
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 the Stirling formula n! 2nnn e-n , show that g(0) 1 n .
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 x p(x) = aE[X] + b. (2) Var[aX + b] = a2 Var[X]. A: Let
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 , if X and Y are independent, prove (1) E[X + Y ] = E[X
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 Bernoulli(p) independently. E[X] = E[Z1 ] + . + E[Zn ]
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: Suppose we have a ve-letter alphabet, A, B, C, D, E, and thei
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 ball again. (1) What is the probability that the secon
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 randomly pick a ball again. (1) What is the probability t
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 . The reason is that all the 2n sequences are equally l
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) = 12/36 = 1/3. p(X = 1, |X Y | > 2) = 3/36 = 1/12. So p(
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: Prove the following two identities: (1) P (A|B) = 1 - P (
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 is the set that corresponds to the event that the numbe
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 is 2? What is its probability? (3) Let Zi = 1 if the i
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 Var[Z ].
(4) Prove Var[X ] = E[X 2 ] E[X ]2 .
Problem
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 person does
not have the disease, then 90% chance the p
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 (B ) = P (Ai )P (B |Ai )/ n=1 P (Aj )P (B |Aj ).
j
Ple
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. Please be precise with notation. Problem 1: Suppose we ip
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 [aX + b] = aE [X ] + b.
Proof.
E [aX + b] =
(aX + b) f (x
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 expectation and variance, thus you cannot just use them right
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 positive. If a person does
not have the disease, then 90% cha
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 ball is blue, we put two blue balls back into the urn. T
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 possible outcomes in the
experiment. A coin has two si