Probability, Random Variables and Density Functions

1. A coin is biased so that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.

2. An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, ¼, ¼, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4pm and 5pm on any sunny Friday. Find the attendant's expected earnings for this particular period.

3. If a dealer's profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function

f(x) = 2(1-x), 0 < x < 1,
0 elsewhere

Find the average profit per automobile.

4. What proportion of individuals can be expected to respond to a certain mail-order solicitation if the proportion X has the density function

f(x) = 2(x+2)/5, 0 < x < 1,
0 elsewhere

5. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2 or 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random variable X representing the number of power failures striking this subdivision.

6. The proportion of people who respond to a certain mail-order solicitation is a random variable X having the density function given in #4. Find the variance of X.

Probability, Random Variables and Density Functions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

1. Given the joint density function for the randomvariables X and Y as
The marginal distribution for the random variable X is
Answer:
2. Given the joint density function for the randomvariables X and Y as
The marginal distribution for the random variable Y is
Answer:
3. The following repr

Let X, Y be independent, standard normal randomvariables, and let U = X + Y and V = X - Y.
(a) Find the joint probability density function of (U, V) and specify its domain.
(b) Find the marginal probability density function of U and V specifying the domain in each case.
(c) Explain why U and V are independent
Joint probab

Suppose X and Y are continuous random variable, X and Y are independent and that x>0 and y>0. The pdf of X is Fx(X) and the pdf of Y is Fy(Y). Find the expressions for the density function of Z in terms of fx and fy, if...
a) Z=X/Y
b) Z=XY

See the attached file.
If X1,X2,..., Xn, are (iid) , from a distribution with mean μ and variance σ^2. Define the sample mean as
Xbar = (X1+X2+...+Xn) / n
(a) Show that the mean and variances of the probability density function of Xbar are given as E(Xbar) = μ
Var(Xbar) = (σ^2)/n
b

Please choose the correct answer and write briefly why.
A property of continuous distributions is that:
a. As with discrete randomvariables, the probability distribution can be approximated by a smooth curve
b. Probabilities for continuous variables can be approximated using discrete randomvariables
c. Unlike discret

I've struggled for 3 days to come up with something approaching a relevant answer but am now desperate.
Could you solve Q3, both a) and b) parts from the Exercise Sheet attached?
Happy to pay 2 credits for both answers.
Thank you very much.
3. The random variable X has an exponential distribution with mean µ. Let Y