Setup: You will use this setup for all problems below.
You observe k i.i.d. copies of the discrete uniform random variable Xi, which takes values 1 through n with equal probability.
Define the random variable M as the maximum of these random variables, M=maxi(Xi).
Problem 1(a)
1 point possible (graded, results hidden)
Find the probability that M≤m, as a function of m, for m∈{1,2,…,n}.
unanswered
Loading
STANDARD NOTATION
Standard Notation
×
Guidelines Example Entries Common mistakes
Symbols These are case sensitive.
Use the correct case as specified in the problem. n and N are different. Do NOT enter x for X
Parentheses Match each open parenthesis with a close parenthesis.
Elementary arithmetic operations Use the symbols +,-,*,/ for addition, subtraction, multiplication, and division, respectively. Enter 1+b*c-d/e for 1+bc−d/e
For multiplication, use * explicitly.
Although the ''pretty'' display underneath your answer looks correct if you do not include * s, your answer will be marked incorrect! Enter b*c for bc in the example above
Enter 2*n*(n+1) for 2n(n+1) Do NOT enter bc for bc
Do NOT enter 2n(n+1) for 2n(n+1)
Exponents Use the symbol ^ to denote exponentiation. Enter 2^n for 2n
Enter x^(n+1) for xn+1
Square root use the string of letters sqrt, followed by enclosing what is under the square root in parentheses. Enter sqrt(-1) for −1−−−√
Mathematical
constants Use the symbol e for the base of the natural logarithm, e.
Use the string of letters pi for π. Enter e^(i*(pi))+1 for eiπ+1
Order of operations 1) parentheses
2) exponents and roots
3) multiplication and division
4) addition and subtraction
When in doubt, use additional parentheses to remove possible ambiguitites. Enter a/b*c for ab⋅c
Enter a/(b*c) for abc
Enter (1/sqrt(2*(pi)))*e^(-(x^2)/2) for 12π−−√e−x22
Do NOT enter a/b*c for abc
Natural logarithm Although in lectures and solved problems we will sometimes use the notation ''log'' (instead of ''ln''), you should use the string of letters ln, followed by the argument enclosed in parentheses. Enter ln(2*x) for ln(2x) Do NOT enter log(2*x) for ln(2x)
Trigonometric
functions Use the usual 3-letter symbols to denote the standard trigonometric functions Enter sin(x) for sin(x) Do NOT enter sin x for sin(x)
Greek letters Use the Latin-character name to denote each Greek letter Enter lambda*e^(-lambda*t) for λe−λt
Enter mu*alpha*theta for muαθ
Factorials, permutations, combinations You will not need enter these for any symbolic answers. Do NOT use ! in your answers as it will not be evaluated correctly!
Close Guide
You have used 0 of 3 attempts Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.
Problem 1(b)
0.5 points possible (graded, results hidden)
Find the probability that M=1.
unanswered
Loading
STANDARD NOTATION
Standard Notation
×
Guidelines Example Entries Common mistakes
Symbols These are case sensitive.
Use the correct case as specified in the problem. n and N are different. Do NOT enter x for X
Parentheses Match each open parenthesis with a close parenthesis.
Elementary arithmetic operations Use the symbols +,-,*,/ for addition, subtraction, multiplication, and division, respectively. Enter 1+b*c-d/e for 1+bc−d/e
For multiplication, use * explicitly.
Although the ''pretty'' display underneath your answer looks correct if you do not include * s, your answer will be marked incorrect! Enter b*c for bc in the example above
Enter 2*n*(n+1) for 2n(n+1) Do NOT enter bc for bc
Do NOT enter 2n(n+1) for 2n(n+1)
Exponents Use the symbol ^ to denote exponentiation. Enter 2^n for 2n
Enter x^(n+1) for xn+1
Square root use the string of letters sqrt, followed by enclosing what is under the square root in parentheses. Enter sqrt(-1) for −1−−−√
Mathematical
constants Use the symbol e for the base of the natural logarithm, e.
Use the string of letters pi for π. Enter e^(i*(pi))+1 for eiπ+1
Order of operations 1) parentheses
2) exponents and roots
3) multiplication and division
4) addition and subtraction
When in doubt, use additional parentheses to remove possible ambiguitites. Enter a/b*c for ab⋅c
Enter a/(b*c) for abc
Enter (1/sqrt(2*(pi)))*e^(-(x^2)/2) for 12π−−√e−x22
Do NOT enter a/b*c for abc
Natural logarithm Although in lectures and solved problems we will sometimes use the notation ''log'' (instead of ''ln''), you should use the string of letters ln, followed by the argument enclosed in parentheses. Enter ln(2*x) for ln(2x) Do NOT enter log(2*x) for ln(2x)
Trigonometric
functions Use the usual 3-letter symbols to denote the standard trigonometric functions Enter sin(x) for sin(x) Do NOT enter sin x for sin(x)
Greek letters Use the Latin-character name to denote each Greek letter Enter lambda*e^(-lambda*t) for λe−λt
Enter mu*alpha*theta for muαθ
Factorials, permutations, combinations You will not need enter these for any symbolic answers. Do NOT use ! in your answers as it will not be evaluated correctly!
Close Guide
You have used 0 of 3 attempts Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.
Problem 1(c)
1 point possible (graded, results hidden)
Find the probability that M=m for m∈{2,3,…n}.
unanswered
Loading
STANDARD NOTATION
Standard Notation
×
Guidelines Example Entries Common mistakes
Symbols These are case sensitive.
Use the correct case as specified in the problem. n and N are different. Do NOT enter x for X
Parentheses Match each open parenthesis with a close parenthesis.
Elementary arithmetic operations Use the symbols +,-,*,/ for addition, subtraction, multiplication, and division, respectively. Enter 1+b*c-d/e for 1+bc−d/e
For multiplication, use * explicitly.
Although the ''pretty'' display underneath your answer looks correct if you do not include * s, your answer will be marked incorrect! Enter b*c for bc in the example above
Enter 2*n*(n+1) for 2n(n+1) Do NOT enter bc for bc
Do NOT enter 2n(n+1) for 2n(n+1)
Exponents Use the symbol ^ to denote exponentiation. Enter 2^n for 2n
Enter x^(n+1) for xn+1
Square root use the string of letters sqrt, followed by enclosing what is under the square root in parentheses. Enter sqrt(-1) for −1−−−√
Mathematical
constants Use the symbol e for the base of the natural logarithm, e.
Use the string of letters pi for π. Enter e^(i*(pi))+1 for eiπ+1
Order of operations 1) parentheses
2) exponents and roots
3) multiplication and division
4) addition and subtraction
When in doubt, use additional parentheses to remove possible ambiguitites. Enter a/b*c for ab⋅c
Enter a/(b*c) for abc
Enter (1/sqrt(2*(pi)))*e^(-(x^2)/2) for 12π−−√e−x22
Do NOT enter a/b*c for abc
Natural logarithm Although in lectures and solved problems we will sometimes use the notation ''log'' (instead of ''ln''), you should use the string of letters ln, followed by the argument enclosed in parentheses. Enter ln(2*x) for ln(2x) Do NOT enter log(2*x) for ln(2x)
Trigonometric
functions Use the usual 3-letter symbols to denote the standard trigonometric functions Enter sin(x) for sin(x) Do NOT enter sin x for sin(x)
Greek letters Use the Latin-character name to denote each Greek letter Enter lambda*e^(-lambda*t) for λe−λt
Enter mu*alpha*theta for muαθ
Factorials, permutations, combinations You will not need enter these for any symbolic answers. Do NOT use ! in your answers as it will not be evaluated correctly!
Close Guide
You have used 0 of 3 attempts Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.
Problem 1(d)
2 points possible (graded, results hidden)
For n=2, find E[M] and Var(M) as a function of k.
E[M]=
unanswered
Loading
Var[M]=
unanswered
To find the probability that M ≤ m for m ∈ {1, 2, ..., n}, we need to consider the probability that each Xi is less than or equal to m. Since each Xi is a discrete uniform random variable that takes values from 1 to n with equal probability, the probability that each Xi is less than or equal to m is given by (m/n)^k, where k is the number of observations.
Therefore, the probability that M ≤ m is (m/n)^k.
Now let's move on to problem 1(b). In this question, we need to find the probability that M = 1. Since M is the maximum of k i.i.d. copies of Xi, for M to be 1, all k observations must be 1. Since each Xi takes values from 1 to n with equal probability, the probability of each observation being 1 is 1/n. Therefore, the probability that M = 1 is (1/n)^k.
For problem 1(c), we need to find the probability that M = m for m ∈ {2, 3, ..., n}. Similar to problem 1(b), for M to be equal to a specific value m, all k observations must be equal to m. Since each Xi takes values from 1 to n with equal probability, the probability of each observation being m is 1/n. Therefore, the probability that M = m is (1/n)^k.
Finally, for problem 1(d), we need to find the expected value (E[M]) and variance (Var[M]) of M when n = 2, as functions of k.
The expected value of M, E[M], is given by the sum of all possible values of M multiplied by their corresponding probabilities. In this case, the possible values of M are 1 and 2, and we already know the probabilities from problems 1(b) and 1(c). So, we can calculate E[M] as follows:
E[M] = 1 * (1/n)^k + 2 * (1/n)^k
The variance of M, Var[M], is a measure of the spread of the random variable M. It is calculated as the sum of the squared differences between each possible value of M and the expected value of M, multiplied by their corresponding probabilities. In this case, we have only two possible values of M (1 and 2), so we can calculate Var[M] as follows:
Var[M] = (1 - E[M])^2 * (1/n)^k + (2 - E[M])^2 * (1/n)^k
Note: The values of E[M] and Var[M] depend on the specific value of k, which is not mentioned in the question.