instructure-uploads.s3.amazonaws.com 3. Suppose we generate a random variable Y

Question

instructure-uploads.s3.amazonaws.com 3. Suppose we generate a random variable Y in the following way. First we flip a fair coin. If the coin is heads, take Y to have a Unif(0, 1) distribution. If the coin is tails, take Y to have a Unif(3, 4) distribution (a) Find the mean of Y (b) Find the standard deviation of Y Hint: Introduce a Bernoulli variable X. Compute E(Y | X-x) and Var(Y | X = x) for x = 0 and 1, what are the random variables E(Y | X) and Var(Y )? Use "iterated expectation" and the "law of total variance". (Review conditional expecta tions by looking at the lecture notes, "All of Statistics" Sec 3.5 or your probability textbook from 36-225)

Explanation / Answer

for first uniform distriution

E(A) =(0+1)/2 =1/2

E(A2)=((0+1)/2)2+((1-0)2/12) =(1/4)+(1/12)=1/3

and for second uniform distribution:

E(B) =(3+4)/2 =7/2

E(B2)=((3+4)/2)2+((4-3)2/12)=(49/4)+(1/12)=148/12=37/3

a)

mean of Y =E(Y)=E(Y|X=heads)*P(X=heads)+E(Y|X=tails)*P(X=tails)

=((0+1)/2)*(1/2)+((3+4)/2)*(1/2)=(1/4)+(7/4)=2

b )

E(Y2)=E(Y2|X=heads)*P(X=heads)+E(Y2|X=tails)*P(X=tails)

=(1/3)*(1/2)+(37/3)*(1/2)=19/3

therefore Variance of Y =Var(Y)=E(Y2)-(E(Y))2 =(19/3)-(2)2 =7/3

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.