Hana Buck
2022-09-23
Answered

Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y ) = 2, and var(X + 2Y ) = 15.

You can still ask an expert for help

Nancy Phillips

Answered 2022-09-24
Author has **12** answers

The correlation coefficient of X and Y is

$${\rho}_{X,Y}=\frac{Cov(X,Y)}{{\sigma}_{X}\ast {\sigma}_{Y}}$$.

By the properties of variance

$$Var(X+2Y)=Var(X)+{2}^{2}Var(Y)+2\ast 1\ast 2\ast Cov(X,Y)$$

$$={\sigma}_{X}^{2}+4{\sigma}_{Y}^{2}+4\rho {\sigma}_{X}{\sigma}_{Y}$$,

and from the fact that

Var(X+2Y)=15

it follows that

$${\sigma}_{X}^{2}+4{\sigma}_{Y}^{2}+4\rho {\sigma}_{X}{\sigma}_{Y}=15$$

$$4+4\ast 2+4\ast \rho \ast \sqrt{4}\sqrt{2}=15$$

which implies that

$$\rho =0.27$$

Result:

$$\rho =0.27$$

$${\rho}_{X,Y}=\frac{Cov(X,Y)}{{\sigma}_{X}\ast {\sigma}_{Y}}$$.

By the properties of variance

$$Var(X+2Y)=Var(X)+{2}^{2}Var(Y)+2\ast 1\ast 2\ast Cov(X,Y)$$

$$={\sigma}_{X}^{2}+4{\sigma}_{Y}^{2}+4\rho {\sigma}_{X}{\sigma}_{Y}$$,

and from the fact that

Var(X+2Y)=15

it follows that

$${\sigma}_{X}^{2}+4{\sigma}_{Y}^{2}+4\rho {\sigma}_{X}{\sigma}_{Y}=15$$

$$4+4\ast 2+4\ast \rho \ast \sqrt{4}\sqrt{2}=15$$

which implies that

$$\rho =0.27$$

Result:

$$\rho =0.27$$

asked 2021-02-21

We wish to estimate what percent of adult residents in a certain county are parents. Out of 600 adult residents sampled, 192 had kids. Based on this, plot a $99\mathrm{\%}$ confidence interval for the proportion of adult residents who are parents in a given county.

Express your answer in the form of three inequalities. Provide your responses in decimal fractions up to three places $<p<$ Express the same answer using a point estimate and a margin of error. Provide your responses as decimals, to three places.

$p=\pm$

asked 2021-03-18

A population of values has a normal distribution with

Find the probability that a single randomly selected value is between 133.6 and 134.1.

Write your answers as numbers accurate to 4 decimal places.

asked 2021-01-02

The amount of time, in minutes, for an airplane to obtain clearance for take off at a certain airport is a random variable Y=3X-2, where X has the density function

$f(x)=\{\begin{array}{l}\frac{1}{4}{e}^{\frac{-x}{4}},x>0\\ 0,\text{else where}\end{array}$

Find the mean and variance of the random variable Y?

asked 2022-09-08

Classify each of the following random variables as discrete or continuous. W = the exact amount of sleep that a randomly selected student from your school got last night.

asked 2022-01-16

What are the mean and standard deviation of a binomial probability distribution with n=169 and $p=\frac{1}{13}$ ?

asked 2022-01-28

How do I calculate the variance and standard deviation of 102, 104, 106, 108, 110?

asked 2021-11-06

If X is a normal random variable with parameters

$\mu =10$

and

${\sigma}^{2}=36$

, compute P[X>5]

and

, compute P[X>5]