A professor pays 25 cents for each blackboard error made in lecture to the student who pointsout the error. In a career ofnyears filled with blackboard errors, the total amount in dollarspaid can be approximated by a Gaussian random variableYnwith expected value 40nandand variance 100n. a. What is the probability that Y20 exceeds 1000?b. How many years n must the professor teach in order that P(Y_n > 1000) > 0.99?

(a) The probability that Y₂₀ exceeds 1000  is 3.91 × 10⁻⁶.

(b) n = 28.09

Step-by-step explanation:

The random variable Yₙ is defined as the total numbers of dollars paid in n years.

It is provided that Yₙ can be approximated by a Gaussian distribution, also known as Normal distribution.

The mean and standard deviation of Yₙ are:

(a)

For n = 20 the mean and standard deviation of Y₂₀ are:

Compute the probability that Y₂₀ exceeds 1000 as follows:

**Use a z table for probability.

Thus, the probability that Y₂₀ exceeds 1000  is 3.91 × 10⁻⁶.

(b)

It is provided that P (Yₙ > 1000) > 0.99.

The value of z for which P (Z < z) = 0.01 is 2.33.

Compute the value of n as follows:

The last equation is a quadratic equation.

The roots of a quadratic equation are:

a = 16

b = -805.4289

c = 10000

On solving the last equation the value of n = 28.09.

step-by-step explanation:

step 1: solve one of the equations for one of the variables. let's solve the first equation for y:

step 2: substitute that equation into the other equation, and solve for x.

step 3: substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.