1. Suppose the population distribution of UCSC GPAs is a normal distribution wit

Question

1. Suppose the population distribution of UCSC GPAs is a normal distribution with mean 3 and variance 0.5. Now suppose you take a sample of UCSC GPAs rounded to the nearest 0.5. GPA 1.5 2.5 3.5 Number of students 10 14 18 20 15 10 4 a) Graph the sample distribution of GPAs and overlay on top of it a graph of the population distribution of GPAs b) What are the sample mean, variance, median, and modes of GPA? 2. Prove that Cov(aX + bY,cZ) -acCov(X, Z) + bcCov(Y, Z) 3. Suppose you have a random variable X distributed as N(3,2). Compute the following probabilities b) P(X

Explanation / Answer

as per chegg policies i am answering 1 question only.

Q3)

x follows N(3,2)

E(X) = 3

V(X) =2

SD(X) = sqrt(2) = 1.414214

a)

P(X>1)

= 1-P(X<1)

I Know that z= (X-mean)/sd

hence, P(X>1) = 1-P(Z<(1-3)/1.414214)

=1-P(Z<-1.41421)

=1-0.07865

0.92135

b)

P(X<12)

I Know that z= (X-mean)/sd

hence, P(X<12) = P(Z<(12-3)/1.414214)

=P(Z<6.363959)

1

c)

P(1<X<12)

P(X<12)-P(X<1)

I Know that z= (X-mean)/sd

hence, P(1<X<12) = P(Z<(12-3)/1.414214) - P(Z<(1-3)/1.414214)

=P(Z<6.363959)-P(Z<-1.41421)

=1-0.07865

0.92135

d)

P(|X-3|<2)

p(X-3)<2 OR P(X-3)>2

THAT IS, P(X<5) or P(X>5)

2*p(z<(5-3)/1.414214)

2*p(z<1.414213)

2*0.92135

1.842700664

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