As another example, consider a random variable distribution with the following probability Y 17 10 -4 -11 P(Y) 0.2 0.3 0.3 0.2 Suppose we take a random sample of size n from this distribution and compute the sample average Sn. What can we say about Sn as the sample size gets bigger and bigger? Well, the LLN says Sn converges to the expected value of the distribution that sample is drawn. Since E[Y] = 3, we can write Σi=1 Yi Sn = N →3 as n→∞ (LLN) In the examples above, we were able to compute the expected value E[Y] of the underlying distribution explicitly using the definition of expected values. But sometimes, Y is a familiar distribution and we might have a formula for E[Y]. In these cases, we can simply use the formula to find E[Y] and the LLN would still mean the sample mean converges to this quantity. In the problems below there are two examples with this flavor. [1] This problem contains two related questions: a) Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming from a binary distribution Y = {0, 1} with P(Y = 1) = 0.7. Find the value that the sample mean S₁ converges as the sample size n increases. b) This question is just a generalization of the previous question. Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming from a binary distribution Y that takes on two values: a and b with probabilities P(Y = a) = p and P(Y = b) = 1 - p. Find the value that the sample mean Sn converges as the sample size n increases.
As another example, consider a random variable distribution with the following probability Y 17 10 -4 -11 P(Y) 0.2 0.3 0.3 0.2 Suppose we take a random sample of size n from this distribution and compute the sample average Sn. What can we say about Sn as the sample size gets bigger and bigger? Well, the LLN says Sn converges to the expected value of the distribution that sample is drawn. Since E[Y] = 3, we can write Σi=1 Yi Sn = N →3 as n→∞ (LLN) In the examples above, we were able to compute the expected value E[Y] of the underlying distribution explicitly using the definition of expected values. But sometimes, Y is a familiar distribution and we might have a formula for E[Y]. In these cases, we can simply use the formula to find E[Y] and the LLN would still mean the sample mean converges to this quantity. In the problems below there are two examples with this flavor. [1] This problem contains two related questions: a) Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming from a binary distribution Y = {0, 1} with P(Y = 1) = 0.7. Find the value that the sample mean S₁ converges as the sample size n increases. b) This question is just a generalization of the previous question. Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming from a binary distribution Y that takes on two values: a and b with probabilities P(Y = a) = p and P(Y = b) = 1 - p. Find the value that the sample mean Sn converges as the sample size n increases.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Show full answers and steps to exercise 1 part a) & b. Explain why and how
Please don’t use R or excel when calculating the results
![As another example, consider a random variable Y with the following probability
distribution
Y 17 10 -4 -11
P(Y) 0.2 0.3 0.3 0.2
Suppose we take a random sample of size n from this distribution and compute the
sample average Sn. What can we say about Sn as the sample size gets bigger and
bigger? Well, the LLN says Sn converges to the expected value of the distribution
that sample is drawn. Since E[Y] =3, we can write
Σ₁=1 Yi
Sn =
N
→ 3
as n→∞ (LLN)
In the examples above, we were able to compute the expected value E[Y] of the
underlying distribution explicitly using the definition of expected values. But
sometimes, Y is a familiar distribution and we might have a formula for E[Y]. In
these cases, we can simply use the formula to find E[Y] and the LLN would still
mean the sample mean converges to this quantity. In the problems below there
are two examples with this flavor.
[1] This problem contains two related questions:
a) Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming
from a binary distribution Y = {0, 1} with P(Y = 1) = 0.7. Find the value
that the sample mean Sn converges as the sample size n increases.
b) This question is just a generalization of the previous question. Suppose
Y₁, Y2, Y3,... be a sequence of independent random variables coming from
a binary distribution Y that takes on two values: a and b with probabilities
P(Y = a) = p and P(Y = b) = 1 - p. Find the value that the sample mean
Sn converges as the sample size n increases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3574180-5be7-46af-9f18-c1fbb9ec682c%2F8a6e7726-ea7e-4abf-8bdd-6e9f42f9db8c%2F1uln26p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:As another example, consider a random variable Y with the following probability
distribution
Y 17 10 -4 -11
P(Y) 0.2 0.3 0.3 0.2
Suppose we take a random sample of size n from this distribution and compute the
sample average Sn. What can we say about Sn as the sample size gets bigger and
bigger? Well, the LLN says Sn converges to the expected value of the distribution
that sample is drawn. Since E[Y] =3, we can write
Σ₁=1 Yi
Sn =
N
→ 3
as n→∞ (LLN)
In the examples above, we were able to compute the expected value E[Y] of the
underlying distribution explicitly using the definition of expected values. But
sometimes, Y is a familiar distribution and we might have a formula for E[Y]. In
these cases, we can simply use the formula to find E[Y] and the LLN would still
mean the sample mean converges to this quantity. In the problems below there
are two examples with this flavor.
[1] This problem contains two related questions:
a) Suppose Y₁, Y2, Y3,... be a sequence of independent random variables coming
from a binary distribution Y = {0, 1} with P(Y = 1) = 0.7. Find the value
that the sample mean Sn converges as the sample size n increases.
b) This question is just a generalization of the previous question. Suppose
Y₁, Y2, Y3,... be a sequence of independent random variables coming from
a binary distribution Y that takes on two values: a and b with probabilities
P(Y = a) = p and P(Y = b) = 1 - p. Find the value that the sample mean
Sn converges as the sample size n increases.
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