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Problem 1. Solve the equation 
via Newton's method,
taking as your initial guess 
. Express your answer to at least
5 decimal places.
[If done correctly, this should only take 4 or 5 iterations. If your
results do not stabilize that fast, you made a mistake somewhere!]
Let 
, 
.
So the answer to 5 decimal places is x= 2.35335.
Problem 2. Consider the differential equation dy/dx = x/y, with the initial condition y(0)=1.
a) Use Euler's method with 
to approximate y(2).
b) Solve the differential equation to find y(2) exactly.
y dy = x dx, so 
. Plugging in y(0)=1 gives C=1/2,
so 
. 
.
Problem 3. Suppose the New York Yankees and the San Diego Padres play a 7 game series. Each game is independent, with the Yankees having a 60% chance of winning. What is the probability that the Yankees will win 4 or more of the 7 games? Your written answer should be correct to at least 3 decimal places (e.g. 0.387, or 38.7%)
This is binomial. n=7, p=0.6, q=1-p=0.4.
.
Problem 4. The rare Tasmanian Guppy (an imaginary fish whose name I just made up) can live for up to a year. Its actual lifespan X (in years) is given by a continuous random variable with pdf
a) Find the corresponding cdf (i.e. F(x)) for all values of x.
b) What is the probability of a randomly selected guppy living between 3 and 6 months?
F(1/2)-F(1/4) = 0.2539.
c) What is the average lifespan of a Tasmanian Guppy.
.
Problem 5. A computer generates seventy-five independent random numbers, each uniformly distributed between 0 and 2.
a) What is the probability that the first random number is between 0.9 and 1.5?
This is uniform with a=0 and b=2. P(0.9 < X < 1.5)=(1.5-0.9)/(2-0)=0.3.
b) Let S be the sum of the random numbers. What is the mean value of S? What is the standard deviation?
For each number, the mean is (a+b)/2=1 and the standard deviation
is 
. For the sum of n=75 numbers, the mean
is 75 times
bigger and the standard deviation 
times bigger, so 
and 
.
c) What is the probability that S is between 70 and 85?
The sum of lots of random variables is normal, in this case with
and 
. Let Z=(S-75)/5. Then
P(70<S<85)=P(-1 < Z< 2)=0.3413+0.4772 = 0.8185.