A liquor store owner is willing to cash personal checks for amounts up to $50, but has become wary of
customers who wear sunglasses. Suppose that 40% of checks written by people wearing sunglasses are
returned by the bank for having insufficient funds, whereas only 5% of checks written by people not
wearing sunglasses are returned for insufficient funds. The owner estimates that 10% of her customers
wear sunglasses.
(a) What is the probability that a randomly selected check is returned by the bank for insufficient
funds?
(b) Suppose that the owner has 32 checks at the end of a busy week. What is the probability that
none of those checks will be returned by the bank for insufficient funds? What are the mean and
standard deviation of the number of checks returned by the bank?
(c) A recently accepted check is returned by the bank for insufficient funds in the checking account.
What is the probability that it was written by someone wearing sunglasses?
Suppose that x has the probability density function
F_(x)(x)=P(x<=x)={(0,x<=0),(1-(1-(x)/(alpha ))^(2),0alpha ):}alpha >0xP(x>(alpha )/(2))xx_(0.5)U=1-(1)/(alpha )xUUx_(1),x_(2),dots,x_(n)x_((n))x( heta , heta +10)Yx=xxeta x∼Uniform( heta , heta +10) and Y|x∼Gamma (x,eta ) heta >0eta >0xY(x,Y)cP(x-Y<=1)Y(f_(Y)(y))xY=y(f_(x|Y)(x|y))U=(x)/(Y)V=Y(U,V)UV
1. A liquor store owner is willing to cash personal checks for amounts up to $50, but has become wary of customers who wear sunglasses. Suppose that 40% of checks written by people wearing sunglasses are returned by the bank for having insufficient funds, whereas only 5% of checks written by people not wearing sunglasses are returned for insufficient funds. The owner estimates that 10% of her customers wear sunglasses.
(a) What is the probability that a randomly selected check is returned by the bank for insufficient funds? [4]
(b) Suppose that the owner has 32 checks at the end of a busy week. What is the probability that none of those checks will be returned by the bank for insufficient funds? What are the mean and standard deviation of the number of checks returned by the bank? [4]
(c) A recently accepted check is returned by the bank for insufficient funds in the checking account. What is the probability that it was written by someone wearing sunglasses? [4]
2. Suppose that X has the probability density function
fxx=-x1{0x}
and distribution function
0 0 >x Fx=PX= {1 (1 2 0<xa > a
for some unknown parameter > 0.
(a) Calculate the mean and variance of X. Set up the integrals needed to obtain these expressions, but feel free to use software capable of symbolic integration to evaluate them. [4] (b) Calculate P(X > /2) as a rational number. [3] (c) Determine the median of X (i.e. the 0.5-quantile, o.5). [3] (d) Let U = 1 X. Use the method of transformations to determine the probability density function of U. Be sure to specify the support of U in your answer. [8] (e) Suppose that X1, X2,...,X, is a random sample from the above probability distribution. Deter- mine the probability density function of the sample maximum, X(n). Simplify your expression for this function, making sure to specify the support. [4] 3. Suppose that X is uniformly distributed over the interval (, + 10) and that the conditional distribution of Y given X = x is gamma with shape parameter x and scale parameter 3:
X Uniform(,+ 10) and Y|X Gamma(X,)
assume that > 0 and 3 > 0. (a) Determine the mean and variance of X.
[4] (b) Determine the mean and variance of Y. (There's a nice route and a very messy route here. Take the nice route.) [6]
4. Let (X, Y) have the joint density
fxYxy=cye-3x1{0<y<x<}
(a) Determine the value of the constant c. This integral should be done by hand.
[4] b) Set up an integral which could be used to calculate P(X -Y 1). There is no need to evaluate the integral. [4]
(c) Determine the marginal density of Y (fy(y)) as well as the conditional density of X given Y = y (fx|(|y)). Be sure to specify the support of both of these distributions. [8] (d) Let U = X/Y and V = Y. Determine the joint pdf of (U, V), making sure to specify the support. [10] (e) Determine whether or not U and V are independent, making sure to justify your reasoning. [2]
