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Problem 1

# During the first year at a university that uses a 4 -point grading system, a freshman took ten 3 -credit courses and received two As, three Bs, four Cs, and one D. a. Compute this student's grade-point average. b. Let the random variable $$X$$ denote the number of points corresponding to a given letter grade. Find the probability distribution of the random variable $$X$$ and compute $$E(X)$$, the expected value of $$X$$.

Expert verified
a. The student's grade-point average is 2.6. b. The probability distribution of the random variable X is: P(X=1) = 1/10, P(X=2) = 4/10, P(X=3) = 3/10, P(X=4) = 2/10, and the expected value of X is 2.6.
See the step by step solution

## Step 1: Compute the student's grade-point average

Recall the 4-point grading system: - A = 4 points - B = 3 points - C = 2 points - D = 1 points The student took ten 3-credit courses and received: - two As (2 × 4 points × 3 credits) - three Bs (3 × 3 points × 3 credits) - four Cs (4 × 2 points × 3 credits) - one D (1 × 1 point × 3 credits) To find the grade-point average, we will add up the total number of grade points earned and divide it by the total number of credits.

## Step 2: Calculate the total grade points and total credits

Total grade points = (2 × 4 × 3) + (3 × 3 × 3) + (4 × 2 × 3) + (1 × 1 × 3) = 24 + 27 + 24 + 3 = 78 Total credits = 10 × 3 = 30

## Step 3: Find the grade-point average

Grade-point average = Total grade points / Total credits = 78 / 30 = 2.6. So, the student's grade-point average is 2.6.

## Step 4: Find the probability distribution of the random variable X

Probability distribution of X: P(X=1) = Probability of getting D = 1/10 P(X=2) = Probability of getting C = 4/10 P(X=3) = Probability of getting B = 3/10 P(X=4) = Probability of getting A = 2/10

## Step 5: Compute the expected value of X

Expected value of X, E(X) = ∑ [x × P(X=x)] E(X) = (1 × 1/10) + (2 × 4/10) + (3 × 3/10) + (4 × 2/10) = 1/10 + 8/10 + 9/10 + 8/10 = 26/10 The expected value of X is E(X) = 2.6. a. The student's grade-point average is 2.6. b. The probability distribution of the random variable X is: P(X=1) = 1/10, P(X=2) = 4/10, P(X=3) = 3/10, P(X=4) = 2/10, and the expected value of X is 2.6.

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