Problem $3$

Let $M$ be a function of $x$ :

$\frac{dM}{dx}=2a+b$

$M(x)=a{x}^2+bx+c$

where $a,b,$ and $c$ are constants. What is the first$-$order derivative of $M$ with respect to $x,$ i$.$e$.\frac{dM}{dx}?$

Problem $4$

Let $M$ be a function of $x$ :

$M(x)=a{x}^2+bx+c$

where $a,b,$ and $c$ are constants. What is the value of $x$ that maximizes the value of $M(x)?$

Problem $5$

A and $B$ are two random variables. Both A and $B$ follow normal distributions as follows:

A follows a normal distribution with a mean of $6$ and a standard deviation of $10,$ i$.$e$.AN(6,10)$

$B$ follows a normal distribution with a mean of $2$ and a standard deviation of $20,$ i$.$e$.BN(2,20)$

What is the mean of the random variable $Z=3A+5B?$

Problem $6$

A and $B$ are two random variables. Both A and $B$ follow normal distributions as follows:

A follows a normal distribution with a mean of $6$ and a standard deviation of $10,$ i$.$e$.AN(6,10)$

$B$ follows a normal distribution with a mean of $2$ and a standard deviation of $20,$ i$.$e$.BN(2,20)$

The covariance between A and $B,$ i$.$e$.,$ cov$(A,B)=7.$

What is the variance of the random variable $Z=3A+5B?$

Problem $7$

$M$ is a random variable following a normal distributions with a mean of $10$ and standard deviation of $20.$ i$.$e$.MN(10,20).$

What distribution does $Q=\frac{M-10}{20}$ follow? What is the mean and standard deviation of $Q?$

Problem $8$

Suppose $T$ is a random variable following standard normal distribution, i$.$e$.,TN(0,1),$ and the probability of $T$ being less than $0\mathrm{.}25$ is $90\%$ i$.$e$.,$

e

Let $M$ be a random variable following a normal distribution with a mean of $10$ and standard deviation of $20,$ i$.$e$..MN(10,20).$

What is the probability of $M$ being less than $5,$ i$.$e