A dealer’s profit, in units of $5000, on a new automobile is given by$Y = X^2$, where X is a random variable having the density function$f(x) = 2(1 − x), 0 \lt x \lt 1$,$f(x)=0$, elsewhere.

(a) Find the probability density function of the random variable Y.

(b) Using the density function of Y, find the probability that the profit on the next new automobile sold by this dealership will be less than$500.

$$\text{ If a dealer’s profit, in units of } \$ \text{5000, on a new automobile can be looked upon as a random variable } X \text{ having the density function }$$

$$f(x)=2(1-x), \text{for } 0<x<1,$$

$$f(x)=0, \text{ elsewhere,}$$

$$\text{ find the average profit per automobile.}$$

What is the dealer’s average profit per automobile if the profit on each automobile is given by $g(X) = X^2$, where X is a random variable having the density function $f(x)=2(1-x)$, for $0 \lt x \lt 1$, $f(x)=0$, elsewhere,

$$\text{Let } X_1 \text{ and } X_2 \text{ be discrete random variables with the joint multinomial distribution }$$

$$f(x_1, x_2) = \binom{2}{x_1, x_2, 2 − x_1 − x_2} \biggr( \frac{1}{4} \biggr)^{x_1} \biggr( \frac{1}{3} \biggr)^{x_2} \biggr( \frac{5}{12} \biggr)^{2-x_1-x_2}$$

$\text{for } x_1 = 0, 1, 2; x_2 = 0, 1, 2; x_1 + x_2 \leq 2; \text{ and zero elsewhere. Find the joint probability distribution of } Y_1 = X_1 + X_2 \text{ and } Y_2 = X_1 − X_2$.

An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by $f(x)=3x^{-4}$, for $x \gt$1, f(x)=0, elsewhere. (a) Verify that this is a valid density function. (b) Evaluate F(x). (c) What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers?

Suppose that X is a random variable with probability distribution fx(x) = 1/4, x = 1, 2, 3, 4. Determine the probability distribution of Y = 2X + 1.

$$\text{Let } X \text{ be a random variable with probability distribution}$$

$$f(x) = \frac{1+x}{2}, −1 < x < 1,$$

$$f(x)=0, \text{ elsewhere.}$$

$$\text{Find the probability distribution of the random variable } Y = X^2.$$

The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x $\leq$ y, and assume that the joint density function of these variables is f(x,y)=2, for 0$\lt x \leq y \lt 1$, f(x,y)=0, elsewhere. (a) Determine if X and Y are independent. (b) Find $P(1/4\lt X \lt 1/2 | Y=3/4).$

The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y=3X−2, where X has the density function $f(x)=\frac{1}{4}e^{-x/4}$, for $x \gt 0$, f(x)=0, elsewhere. Find the mean and variance of the random variable Y.

Show that f(x, y)=1/x, 0<y<x<1, is a joint density function. Assuming that f is the joint density function of X,Y, find (a) the marginal density of Y; (b) the marginal density of X; (c) E[X]; (c) E[Y].

In a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each. The data are as follows: Company A: 9.3 8.8 6.8 8.7 8.5 6.7 8.0 6.5 9.2 7.0 Company B: 11.0 9.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6. Compute both the mean and the variance in “flexibility” for both company A and company B. Does there appear to be a difference in flexibility between company A and company B?

In a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each. The data are as follows: Company A: 9.3 8.8 6.8 8.7 8.5 6.7 8.0 6.5 9.2 7.0 Company B: 11.0 9.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6 (a) Calculate the sample mean and median for the data for the two companies. (b) Plot the data for the two companies on the same line and give your impression regarding any apparent differences between the two companies.

How many times does a typical person blink her eyes in a lifetime?

$$\text{Given the random variable } X \text{ with probability distribution}$$

$$f(x) =2x, 0 < x < 1,$$

$$f(x)=0, \text{ elsewhere,}$$

$$\text{ find the probability distribution of } Y = 8X^3.$$