For each of the following functions, (i) find the constant c so that f(x) is a pdf of a

Nicol (see References) lets the pdf of \(X\) be defined by \(f(x)= \begin{cases}x, & 0 \leq x \leq 1 \\ c / x^{3}, & 1 \leq x<\infty \\ 0, & \text { elsewhere }\end{cases}\) Find (a) The value of \(c\) so that \(f(x)\) is a pdf. (b) The mean of \(X\) (if it exists). (c) The variance of \(X\) (if it exists). (d) \(P(1 / 2 \leq X \leq 2)\). Equation Transcription: { Text Transcription: X f(x)= { 0, otherwise c/x^3,x 1 x<0 x 1 f(X) P(½ < or = X < or = 2)

The pdf of X is \(f(x) = c/x^2, 1 < x < \infty\). (a) Calculate the value of c so that f(x) is a pdf. (b) Show that E(X) is not finite.

Sketch the graphs of the following pdfs and find and sketch the graphs of the cdfs associated with these distributions (note carefully the relationship between the shape of the graph of the pdf and the concavity of the graph of the cdf): (a) \(f(x)=\left(\frac{3}{2}\right) x^{2}, \quad-1<x<1\). (b) \(f(x)=\frac{1}{2}, \quad-1<x<1\). (c) \(f(x)= \begin{cases}x+1, & -1<x<0, \\ 1-x, & 0 \leq x<1 .\end{cases}\) Equation Transcription: { Text Transcription: f(x)=(3/2)x^2, -1<x<1 f(x)=12, -1<x<1 f(x)= {_1-x, 0 x < 1 ^x+1 -1< x< 0

Problem 11E The pdf of Y is g(y) = d/y3, 1 < y < ?. (a) Calculate the value of d so that g(y) is a pdf. (b) Find E(Y). (c) Show that Var(Y) is not finite.

Problem 13E The logistic distribution is associated with the cdf F(x) = (1 + e ?x ) ?1, ?? < x < ?. Find the pdf of the logistic distribution and show that its graph is symmetric about x = 0.

Problem 9E Let the random variable X have the pdf f (x) =2(1 ? x), 0 ? x ? 1, zero elsewhere. (a) Sketch the graph of this pdf. (b) Determine and sketch the graph of the cdf of X. (c) Find (i) P(0 ? X ? 1/2), (ii) P(1/4 ? X ? 3/4), (iii) P(X = 3/4), and (iv) P(X ? 3/4).

Problem 14E Let f (x) = 1/2, 0 < x < 1 or 2 < x < 3, zero elsewhere, be the pdf of X. (a) Sketch the graph of this pdf. (b) Define the cdf of X and sketch its graph. (c) Find q1 = ?0.25. (d) Find m = ?0.50. Is it unique? (e) Find q3 = ?0.75.

The life \(X\) (in years) of a voltage regulator of a car has the pdf \(f(x)=\frac{3 x^{2}}{7^{3}} e^{-(x / 7)^{3}}, \quad 0<x<\infty\). (a) What is the probability that this regulator will last at least 7 years? (b) Given that it has lasted at least 7 years, what is the conditional probability that it will last at least another 3.5 years? Equation Transcription: Text Transcription: X f(x)=3x^2/7^3 e^-(x/7)^3, 0 < x< infinity

Problem 16E Let f (x) = (x + 1)/2, ?1 < x < 1. Find (a) ?0.64, (b) q1 = ?0.25, and (c) ?0.81.

An insurance agent receives a bonus if the loss ratio \(L\) on his business is less than \(0.5\), where \(L\) is the total losses (say, \(X\) ) divided by the total premiums (say, \(T\) ). The bonus equals \((0.5-L)(T / 30)\) if \(L<0.5\) and equals zero otherwise. If \(X\) (in $100,000 ) has the pdf \(f(x)=\frac{3}{x^{4}}, \quad x>1\) and if \(T\) (in $100,000 ) equals 3 , determine the expected value of the bonus. Equation Transcription: Text Transcription: X L 0.5 T (0.5-L)(T/30) L<0.5 f(x)=3/x^4, x>1

The weekly demand \(X\) for propane gas (in thousands of gallons) has the pdf \(f(x)=4 x^{3} e^{-x^{4}}, \quad 0<x<\infty\). If the stockpile consists of two thousand gallons at the beginning of each week (and nothing extra is received during the week), what is the probability of not being able to meet the demand during a given week? Equation Transcription: Text Transcription: X f(x)=4x^3 e^-x4, 0<x<infinity

Problem 19E The total amount of medical claims (in $100,000) of the employees of a company has the pdf that is given by f (x) = 30x(1 ? x)4, 0 < x < 1. Find (a) The mean and the standard deviation of the total in dollars. (b) The probability that the total exceeds $20,000.

Show that the mean, variance, and mgf of the uniform distribution are as given in this section.

Let f(x) = 1/2, 1 x 1, be the pdf of X. Graph the pdf and cdf, and record the mean and variance of X.

Customers arrive randomly at a bank tellers window. Given that one customer arrived during a particular 10-minute period, let X equal the time within the 10 minutes that the customer arrived. If X is U(0, 10), find (a) The pdf of X. (b) P(X 8). (c) P(2 X < 8). (d) E(X). (e) Var(X).

If the mgf of X is M(t) = e5t e4t t , t = 0, and M(0) = 1, find (a) E(X), (b) Var(X), and (c) P(4.2 < X 4.7).

Let Y have a uniform distribution U(0, 1), and let W = a + (b a)Y, a < b. (a) Find the cdf of W. Hint: Find P[a + (b a)Y w]. (b) How is W distributed?

A grocery store has n watermelons to sell and makes $1.00 on each sale. Say the number of consumers of these watermelons is a random variable with a distribution that can be approximated by f(x) = 1 200 , 0 < x < 200, a pdf of the continuous type. If the grocer does not have enough watermelons to sell to all consumers, she figures that she loses $5.00 in goodwill from each unhappy customer. But if she has surplus watermelons, she loses 50 cents on each extra watermelon. What should n be to maximize profit? Hint: If X n, then her profit is (1.00)X + (0.50)(n X); but if X > n, her profit is (1.00)n+(5.00)(Xn). Find the expected value of profit as a function of n, and then select n to maximize that function.

For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable X, (ii) find the cdf, F(x) = P(X x), (iii) sketch graphs of the pdf f(x) and the cdf F(x), and (iv) find and 2: (a) f(x) = 4xc, 0 x 1. (b) f(x) = c x, 0 x 4. (c) f(x) = c/x3/4, 0 < x < 1.

For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable X, (ii) find the cdf, F(x) = P(X x), (iii) sketch graphs of the pdf f(x) and the distribution function F(x), and (iv) find and 2: (a) f(x) = x3/4, 0 < x < c. (b) f(x) = (3/16)x2, c < x < c. (c) f(x) = c/ x, 0 < x < 1. Is this pdf bounded?

Let the random variable X have the pdf f(x) = 2(1 x), 0 x 1, zero elsewhere. (a) Sketch the graph of this pdf. (b) Determine and sketch the graph of the cdf of X. (c) Find (i) P(0 X 1/2), (ii) P(1/4 X 3/4), (iii) P(X = 3/4), and (iv) P(X 3/4).

The pdf of X is f(x) = c/x2, 1 < x < . (a) Calculate the value of c so that f(x) is a pdf. (b) Show that E(X) is not finite.

The pdf of Y is \(g(y) = d/y^3, 1 < y < \infty\). (a) Calculate the value of d so that g(y) is a pdf. (b) Find E(Y). (c) Show that Var(Y) is not finite.

Sketch the graphs of the following pdfs and find and sketch the graphs of the cdfs associated with these distributions (note carefully the relationship between the shape of the graph of the pdf and the concavity of the graph of the cdf): (a) f(x) = 3 2  x2, 1 < x < 1. (b) f(x) = 1 2 , 1 < x < 1. (c) f(x) = ! x + 1, 1 < x < 0, 1 x, 0 x < 1.

The logistic distribution is associated with the cdf F(x) = (1 + ex)1, < x < . Find the pdf of the logistic distribution and show that its graph is symmetric about x = 0.

Let f(x) = 1/2, 0 < x < 1 or 2 < x < 3, zero elsewhere, be the pdf of X. (a) Sketch the graph of this pdf. (b) Define the cdf of X and sketch its graph. (c) Find q1 = 0.25. (d) Find m = 0.50. Is it unique? (e) Find q3 = 0.75.

The life X (in years) of a voltage regulator of a car has the pdf f(x) = 3x2 73 e(x/7)3 , 0 < x < . (a) What is the probability that this regulator will last at least 7 years? (b) Given that it has lasted at least 7 years, what is the conditional probability that it will last at least another 3.5 years?

Let f(x) = (x + 1)/2, ?1 < x < 1. Find (a) \(\pi_{0.64}\), (b) \(q_1 = \pi_{0.25}\), and (c) \(\pi_{0.81}\).

An insurance agent receives a bonus if the loss ratio L on his business is less than 0.5, where L is the total losses (say, X) divided by the total premiums (say, T). The bonus equals (0.5 ? L)(T/30) if L < 0.5 and equals zero otherwise. If X (in $100,000) has the pdf \(f(x)=\frac{3}{x^4}\), x > 1, and if T (in $100,000) equals 3, determine the expected value of the bonus.

The weekly demand X for propane gas (in thousands of gallons) has the pdf f(x) = 4x3ex4 , 0 < x < . If the stockpile consists of two thousand gallons at the beginning of each week (and nothing extra is received during the week), what is the probability of not being able to meet the demand during a given week?

The total amount of medical claims (in $100,000) of the employees of a company has the pdf that is given by f(x) = 30x(1 x)4, 0 < x < 1. Find (a) The mean and the standard deviation of the total in dollars. (b) The probability that the total exceeds $20,000.

Nicol (see References) lets the pdf of X be defined by f(x) = x, 0 x 1, c/x3, 1 x < , 0, elsewhere. Find (a) The value of c so that f(x) is a pdf. (b) The mean of X (if it exists). (c) The variance of X (if it exists). (d) P(1/2 X 2).

Let X1, X2, ... , Xk be random variables of the continuous type, and let f1(x),f2(x), ... ,fk(x) be their corresponding pdfs, each with sample space S = (,). Also, let c1, c2, ... , ck be nonnegative constants such that k i=1 ci = 1. (a) Show that k i=1 cifi(x) is a pdf of a continuous-type random variable on S. (b) If X is a continuous-type random variable with pdf k i=1 cifi(x) on S, E(Xi) = i, and Var(Xi) = 2 i for i = 1, ... , k, find the mean and the variance of X.