Let ?1,?2,…,??be a collection of independent discrete random variables that all take the value 1 with...
Let ?1,?2,…,??be a collection of independent discrete random
variables that all take the value 1 with probability p and take the
value 0 with probability (1-p).
a) Compute the mean and the variance of ?1 (which is the same for
?2, ?3, etc.)
b) Use your answer to (a) to compute the mean and variance of ?̂ = 1/n (?1 + ?2 + ⋯+ ??), which is the proportion of “ones” observed in the n instances of ??.
c) Suppose n = 10,000. Use Chebyshev’s inequality to provide an upper bound for the probability that the difference between ?̂ and ? exceeds 0.05.
d) Use calculus to show that if ? is a number between 0 and 1, then ?(1 − ?) ≤ 1/4 .
e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on ?, for the probability that the difference between ?̂ and ? exceeds 0.05.
f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking them a yes-no question, and using the result to make an inference about the whole population.
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Let ?1,?2,…,??be a collection of independent discrete random
variables that all take the value 1 with...
Let X and Y be two independent and identically distributed random variables with expected value 1...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
Let X1, ..., X20 be independent Poisson random variables with mean one. 1 2 (a) Use...
Let X1, ..., X20 be independent Poisson random variables with mean one. 1 2 (a) Use the Markov inequality to obtain a bound on P (X1 + X2 + · · · + X20 > 15). (b) Use the central limit theorem to approximate the above probability.
Let X1, · · · , X20 be independent Poisson random variables with mean (print please)...
Let X1, · · · , X20 be independent Poisson random variables with mean (print please) 1. (i) Use the pmf of Poisson distribution to compute P(X1 + · · · + X20 > 15). (ii) Use the Markov inequality to obtain a bound on P(X1 + · · · + X20 > 15). (iii) Use the central limit theorem to approximate P(X1 + · · · + X20 > 15).
Let Mn be the maximum of n independent U(0, 1) random variables. a. Derive the exact...
Let Mn be the maximum of n independent U(0, 1) random variables. a. Derive the exact expression for P(|Mn − 1| > ε). Hint: see Section 8.4. b. Show that limn→∞ P(|Mn − 1| > ε) = 0. Can this be derived from Chebyshev’s inequality or the law of large numbers?
LU 22 2. We know that the sample variance follows a chi-square distribution: Sanx?(n-1). (a) (5...
LU 22 2. We know that the sample variance follows a chi-square distribution: Sanx?(n-1). (a) (5 points) Use this fact to show that E(S) = 02. (Hint: Find the mean of the x as then mean of a Gamma distribution.) (b) (5 points) Use Markov's inequality to find an upper bound on the probability that the sample variance is twice the true variance, i.e. P(S? > 20%).
1. There are times when a shifted exponential model is appropriate. That is, let the pdf of X be ...
1. There are times when a shifted exponential model is appropriate. That is, let the pdf of X be (a) Find the cdf of X. (b) Find the mean and variance of X. 2. Suppose X is a Gamma random variable with pdf 「(a)go Show that the moment generating function is M(t) 3, Let X equal the nurnber out of n 48 mature aster seeds that will germinate when p- 0.75 is the probability that a particular seed germinates. Approximate...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
you take a random sample size of 1500 from population 1 and a random sample size...
you take a random sample size of 1500 from population 1 and a random sample size of 1500 from population two. the mean of the first sample size is 76; the sample standard deviation is 20. the mean of the second sample is 62; the sample standard deviation is 18. construct the 90% confidence interval estimate of the difference between the means of the two populations representwd here and report both the upper and lower bound of the interval.
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables,...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
Let Y1, Y2, , Yn be independent, normal random variables, each with mean μ and variance...
Let Y1, Y2, , Yn be independent, normal random variables, each
with mean μ and variance σ^2.
(a) Find the density function of
f Y(u) =
(b) If σ^2 = 25 and n = 9, what is the
probability that the sample mean, Y, takes on a value that is
within one unit of the population mean, μ?
That is, find P(|Y − μ| ≤ 1). (Round your answer to four decimal
places.)
P(|Y − μ| ≤ 1) =
(c)...
LU 22 2. We know that the sample variance follows a chi-square distribution: Sanx?(n-1). (a) (5 points) Use this fact to show that E(S) = 02. (Hint: Find the mean of the x as then mean of a Gamma distribution.) (b) (5 points) Use Markov's inequality to find an upper bound on the probability that the sample variance is twice the true variance, i.e. P(S? > 20%).
1. There are times when a shifted exponential model is appropriate. That is, let the pdf of X be (a) Find the cdf of X. (b) Find the mean and variance of X. 2. Suppose X is a Gamma random variable with pdf 「(a)go Show that the moment generating function is M(t) 3, Let X equal the nurnber out of n 48 mature aster seeds that will germinate when p- 0.75 is the probability that a particular seed germinates. Approximate...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
Let Y1, Y2, , Yn be independent, normal random variables, each
with mean μ and variance σ^2.
(a) Find the density function of
f Y(u) =
(b) If σ^2 = 25 and n = 9, what is the
probability that the sample mean, Y, takes on a value that is
within one unit of the population mean, μ?
That is, find P(|Y − μ| ≤ 1). (Round your answer to four decimal
places.)
P(|Y − μ| ≤ 1) =
(c)...
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