Question

Question Help Compute the following probabilities: If Y is distributed N (-2.4). Pr (Ys-5)(Round your response to four decimal places.)

Answer 1

a) The marginal distribution of Y is given by the column totals in the table

that is P(Y=0)=0.068 and P(Y=1)=0.932

The expected value of Y is

$\begin{align*} E(Y)&=\sum_{y=0,1}yP(y)\\ &=0\times P(Y=0)+1\times P(Y=1)\\ &=0\times 0.068+1\times 0.932\\ &=0.932 \end{align*}$

b) From the column totals, we know that if the total labor force is 1, then 0.068 is unemployed. Since unemployment rate is the fraction of the labor force that is unemployed,
Unemployment rate is 0.068

E(Y)=0.932

1-E(Y)=1-0.932 = 0.068

Hence the unemployment rate is given by 1-E(Y)

c) We know the following probabilities

Marginal probabilities of X

P(X=0) = 0.639

P(X=1)=0.361

The joint probabilities

P(X=0,Y=0) = 0.053

P(X=0,Y=1) = 0.586

P(X=1,Y=0) = 0.015

P(X=1,Y=1) = 0.346

The conditional distribution of Y given X=1 is

$\begin{align*} &\text{using the formula for conditional distribution}\\ &P(Y=0\mid X=1)=\frac{P(X=1,Y=0)}{P(X=1)}=\frac{0.015}{0.361}=0.0416 \\ &P(Y=1\mid X=1)=\frac{P(X=1,Y=1)}{P(X=1)}=\frac{0.346}{0.361}=0.9584\\ \end{align*}$

The conditional expectation of Y given X=1 is

$\begin{align*} E(Y\mid X=1)&=\sum_{y=0,1}yP(y\mid X=1)\\ &=0\times P(Y=0\mid X=1)+1\times P(Y=1\mid X=1)\\ &=0\times 0.0416+1\times 0.9584\\ &=0.958 \end{align*}$

The conditional distribution of Y given X=0 is

$\begin{align*} &\text{using the formula for conditional distribution}\\ &P(Y=0\mid X=0)=\frac{P(X=0,Y=0)}{P(X=0)}=\frac{0.053}{0.639}=0.0829 \\ &P(Y=1\mid X=0)=\frac{P(X=0,Y=1)}{P(X=0)}=\frac{0.586}{0.639}=0.9171\\ \end{align*}$

The conditional expectation of Y given X=0 is

$\begin{align*} E(Y\mid X=0)&=\sum_{y=0,1}yP(y\mid X=0)\\ &=0\times P(Y=0\mid X=0)+1\times P(Y=1\mid X=0)\\ &=0\times 0.0829+1\times 0.9171\\ &=0.917 \end{align*}$

d). Using the result from part b), we know that unemployment rate is given by 1-E(Y)

i) Unemployment rate for college graduates (X=1) is 1-E(Y|X=1) =1-0.958=0.042

ii) Unemployment rate for non-college graduates (X=0) is 1-E(Y|X=0) =1-0.917=0.083

e) Y=0 is the event that a randomly selected member of this population reports being unemployed.

Let X=1 is the event that a randomly selected member of this population is a college graduate.

Let X=0 is the event that a randomly selected member of this population is a non-college graduate.

the probability that  a randomly selected member of this population is a college graduate (X=1) given that this worker is unemployed (Y=0) is given by

$\begin{align*} P(X=1\mid Y=0)&=\frac{P(X=1\cap Y=0)}{P(Y=0)}\quad\text{using the formula for conditional distribution}\\ &=\frac{0.015}{0.068}\\ &=0.221 \end{align*}$

ans: the probability that  a randomly selected member of this population is a college graduate given that this worker is unemployed is 0.221

the probability that  a randomly selected member of this population is a non-college graduate (X=0) given that this worker is unemployed (Y=0) is given by

$\begin{align*} P(X=0\mid Y=0)&=\frac{P(X=0\cap Y=0)}{P(Y=0)}\quad\text{using the formula for conditional distribution}\\ &=\frac{0.053}{0.068}\\ &=0.779 \end{align*}$

ans: the probability that  a randomly selected member of this population is a non-college graduate given that this worker is unemployed is 0.779

f) 2 events A and B are independent if their joint probability is equal to the product of marginal probability.

That is A and B are independent if

$\begin{align*} P(A\cap B)=P(A)\times P(B) \end{align*}$

We will pick the first cell, the corresponding events are

X=0: selected member of this population is a non-college graduate

Y=0: selected member of this population is a unemployed

The joint probability of X=0 and Y=0 is

$\begin{align*} P(X=0,Y=0)=0.053 \end{align*}$

The marginal probability of X=0 is P(X=0)=0.639

The marginal probability of Y=0 is P(Y=0)=0.068

the product is

$\begin{align*} P(X=0)\times P(Y=0)=0.639\times 0.068=0.043 \end{align*}$

This is not equal to the joint probability.

Hence X=0 and Y=0 are not independent.

We can check this for others, but one is sufficient.

We can say that educational achievement and employment status are not independent.

That is employment status depends on educational achievement.