Compare your answers to Exercises $5$ and $6$. Is Euler's method doing a good job in this case? What would you do to avoid the difficulties that arise in this case?

What can you say about solutions of $d y / d t=-y^2$ for which the initial condition $y(0)$ satisfies the inequality $-1<y(0)<-1 / 2$ ? [Hint: You could find the general solution, but what information can you get from your answer to previous part alone?]

We have seen that if Y has a binomial distribution with parameters n and p, then

$$Y / n$$

is an unbiased estimator of p. To estimate the variance of Y , we generally use

$$n ( Y / n ) ( 1 - Y / n )$$

. Modify

$$n ( Y / n ) ( 1 - Y / n )$$

slightly to form an unbiased estimator of V(Y ).

We have seen that if Y has a binomial distribution with parameters n and p, then Y/n is an unbiased estimator of p. To estimate the variance of Y, we generally use

$$n ( Y / n ) ( 1 - Y / n )$$

. Show that the suggested estimator is a biased estimator of V(Y ).

Give one possible sample of size 4 from each of the following populations: a. All daily newspapers published in the United States b. All companies listed on the New York Stock Exchange c. All students at your college or university d. All grade point averages of students at your college or university

The number of breakdowns per week for a type of minicomputer is a random variable Y with a Poisson distribution and mean

$$\lambda$$

. A random sample

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

of observations on the weekly number of breakdowns is available. The weekly cost of repairing these breakdowns is

$$C = 3 Y + Y ^ { 2 }$$

. Show that

$$E ( C ) = 4 \lambda + \lambda ^ { 2 }$$

.

The number of breakdowns per week for a type of minicomputer is a random variable Y with a Poisson distribution and mean

$$\lambda$$

. A random sample

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

of observations on the weekly number of breakdowns is available. Find a function of

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

that is an unbiased estimator of E(C). [Hint: Use what you know about

$$\overline { Y } \text { and } ( \overline { Y } ) ^ { 2 }$$

.]

The number of breakdowns per week for a type of minicomputer is a random variable Y with a Poisson distribution and mean

$$\lambda$$

. A random sample

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

of observations on the weekly number of breakdowns is available. Suggest an unbiased estimator for

$$\lambda$$

.

Consider the differential equation

$$\frac{d y}{d t}=\frac{y}{t^2} .$$

Why doesn't this example contradict the Uniqueness Theorem?

Use technology to sketch the level curve of f(x, y)=4x-2y+3 that passes through P(1, 2) and draw the gradient vector at P.

Sketch the level curve of f(x, y) that passes through P and draw the gradient vector at P. f(x, y)=4x-2y+3; P(1, 2)

Consider the differential equation

$$\frac{d y}{d t}=\frac{y}{t^2} .$$

Show that there are infinitely many other functions that satisfy the differential equation, that agree with this solution when $t \leq 0$, but that are nonzero when $t>0$. [Hint: You need to define these functions using language like " $y(t)=\ldots$ when $t \leq 0$ and $y(t)=\ldots$ when $t>0 . "]$

Solve the given initial-value problem.

$$\frac{d y}{d t}-2 y=7 e^{2 t}, \quad y(0)=3$$

Find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.)

$$\frac{d y}{d t}=\frac{t}{t^2y+y}$$