2. A random test score X is obtained from a class with students who fall into two groups. For the first group X is conditionally Gaussian with mean 80 and variance 20, while for the second group X is conditionally Gaussian with mean 20 and variance 10. The probability that a student is in the first group is 0.6. (a) Find E [X] and Var [X] . (b) Assuming the grades are assigned as in the notes, find the probability that a randomly selected exam gets an A, B, C, D or F grade (in terms of the Φ function

E (X) = 56 and V (X) = 880.

Step-by-step explanation:

The random variable X denotes the test scores obtained from a class with students who fall into two groups.

Let's denote the test scores of first group as X₁ and the test scores of second group as X₂.

The information provided is:

The probability of selecting a student from the first group is, p = 0.60.

Then the probability of selecting a student from the second group is,

q = 1 - p = 1 - 0.60 = 0.40.

(a)

Compute the expected test score obtained as follows:

E (X) = p × E (X₁) + q × E (X₂)

Thus, the expected test score obtained is E (X) = 56.

Compute the value of E (X₁²) as follows:

Compute the value of E (X₂²) as follows:

Compute the value of E (X²) as follows:

Compute the variance of the test scores obtained as follows:

Thus, the variance of the test scores obtained is, V(X) = 880.

(b)

Since the division of grades is not provided, i.e. which score is assigned what grade we cannot compute the probability of randomly selecting an exam with grade A, B, C, D or F.

the answer is 0.071

good luck! : d

69.64%

step-by-step explanation: