After the final exam, the professor has curved the grade to approximately Gaussian distribution with mean 80 and variance 25. Suppose there are N students in this class. How many students have score at least 70? (Let (t) be the cumulative distribution function of N(0, 1), standard Gaussian random variable. You can leave the final result containing D.) (Note: Don't forget continuous correction!)

X and Y be standard normal random variables with correlation p. Compute the joint and marginal distributions of X+Y and X-Y. Are X+Y and X−Y independent?

Suppose X is a random variable of a normal distribution with E(X)=2 and Var(X)=4. Find P(X<4)

Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)

Suppose a random variable X has the following pdf.   The random variable X has a Beta distribution with α = 3 and β = 2.  Suppose we wanted to determine Var( 10X - 3 ).   By the rules for variances, Var( 10X - 3 ) =  * Var(X) +    where, using the fact that X has a Beta distribution, Var(X) =  Express answer as a decimal.  Do not round.

Please help with this practice problem

Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1:  Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2:  Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?

How do you find the mean and variance of part(d)?I need the detailed solution.

Please answer both parts of the question thanks

Which of the following statement is INCORRECT? If Y, follows the standard normal distribution and Y, follows a chis-square distribution with degrees of freedom n, then nY/Y t distribution. If X1,.., Xn is a random sample from a normal population with population mean 1 and variance 1, then (X -1) + (X - 1 (X, - 1) follows a chi-square distribution with degrees of freedom n. If X,..., Xn is a random sample from a normal distribution with population mean 0 and a known variance o, then must follow a distribution, where X denotes the sample mean and S denotes the sample standard deviation. If T follows at distribution with degrees of freedom n, then T must follow an F distribution with degrees of freedom 1 and n.