Random Variable Problems & Solutions

Use the one-standard-deviation $x^{2}$ -test and the one-standard-deviation $x^{2}$ -interval procedure to conduct the required hypothesis test and obtain the specified confidence interval. $s = 5$ and $n = 20$
a. $H_{0} : σ = 6, H_{a} : σ \neq 6, α = 0.05$ .
b. 95% confidence interval

shadsiei 2021-05-19

Random variables $X_{1}, X_{2}, . . ., X_{n}$ are independent and identically distributed. 0 is a parameter of their distribution.
State the definition for a pivotal function of X, the vector of random variables, and 0.

Ernstfalld 2021-05-18

Find the expected values and variances of the exponential random variables with the density functions given below
$3 e^{- 3 x}$

banganX 2021-05-18

Two random variables X and Y with joint density function given by:
$f (x, y) = {\begin{cases} \frac{1}{3} (2 x + 4 y) & 0 \leq x, \leq 1 \\ 0 & e l s e w h e r e \end{cases}$
Find $P (x < \frac{1}{3})$

Haven 2021-05-18

Random variables Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: X-20

Cabiolab 2021-05-17

The yield strength of CP titanium welds was measured for welds cooled at rates of $10^{\circ} C; s, 15^{\circ} C; s, and 28^{\circ} C; s$ . The results are presented in the following table. (Based on the article “Advances in Oxygen Equivalence Equations for Predicting the Properties of Titanium Welds,” D. Harwig, W. Ittiwattana, and H. Castner, The Welding Journal, 2001:126s-136s.)

$\begin{matrix} Cooling Rate & Yiled Strengths \\ 10 & 71.00 & 75.00 & 79.67 & 81.00 & 75.50 & 72.50 & 73.50 & 78.50 & 78.50 \\ 15 & 63.00 & 68.00 & 73.00 & 76.00 & 79.67 & 81.00 \\ 28 & 68.65 & 73.70 & 78.40 & 84.40 & 91.20 & 87.15 & 77.20 & 80.70 & 84.85 & 88.40 \end{matrix}$
a. Construct an ANOVA table. You may give a range for the P-value. b. Can you conclude that the yield strength of CP titanium welds varies with the cooling rate?

Ernstfalld 2021-05-17

Suppose the random variables X and Y have a pdf given by
f(x,y)=x+y on 0 Find F( $\frac{1}{4}, \frac{1}{2}) .$

allhvasstH 2021-05-16

Let $X_{1} . . . ., X_{n} a n d Y_{1}, . . ., Y_{m}$ be two sets of random variables. Let $a_{i}, b_{j}$ be arbitrary constant.
Show that
$C o v (\sum_{i = 1}^{n} a_{i} X_{i}, \sum_{j = 1}^{m} b_{j} Y_{j}) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{i} b_{j} C o v (X_{i}, Y_{j})$

e1s2kat26 2021-05-16

Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma. Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was $x = 5.35 m \frac{g}{d l}$ . The distribution of uric acid in healthy adult males can be assumed to be normal with $σ = 1.85 m \frac{g}{d l} σ = 1.85 m \frac{g}{d l}$ . What conditions are necessary for your calculations?

Mylo O'Moore 2021-05-16

X and Y are independent rv's with moment generating functions $M_{X} (t)$ and $M_{Y} (t)$ , respectively. If Z=X+Y, we need to show that $M_{Z} (t) = M_{X} (t) M_{Y} (t)$ .

Ayaana Buck 2021-05-13

Let the continuous random variables X and Y have joint pdf

$f (x, y) = e^{- x - y}, 0$

a) $e^{- y}$

b) $e^{- x}$

c) $\frac{e^{y}}{e^{- x}}$

he298c 2021-05-13

A boy is to sell lemonade to make some money to afford some holiday shopping. The capacity of the lemonade bucket is 1. At the start of each day, the amount of lemonade in the bucket is a random variable X, from which a random variable Y is sold during the day. The two random variables X and Y are jointly uniform.
Find and sketch the CDF and the pdf of Z which is the amount of lemonade remaining at the end of the day. Clearly indicate the range of Z

boitshupoO 2021-05-12

Let $X_{1}, X_{2}, . . ., X_{6}$ be an i.i.d. random sample where each $X_{i}$ is a continuous random variable with probability density function
$f (x) = e^{- (x - 0)}, x > 0$
Find the probability density function for $X_{6}$ .

ruigE 2021-05-08

Two random variables X and Y with joint density function given by:
$f (x, y) = {\begin{cases} \frac{1}{3} (2 x + 4 y) & 0 \leq x, \leq 1 \\ 0 & e l s e w h e r e \end{cases}$
Find the marginal density of Y.

Jaden Easton 2021-05-08

The two-way table summarizes data from an experiment comparing the effectiveness of three different diets (A, B, and C) on weight loss. Researchers randomly assigned 300 volunteer subjects to the three diets. The response variable was whether each subject lost weight over a 1-year period.

$Diet Lost weight? \begin{array}{lcccc} A & B & C & Total \\ Yes & 60 & 180 \\ No & 40 & 120 \\ Total & 90 & 100 & 110 & 300 \end{array}$

Suppose we randomly select one of the subjects from the experiment. Show that the events "Diet B" and "Lost weight" are independent.

Brennan Flores 2021-05-05

Suppose the random variables X and Y have a pdf given by
f(x,y)=x+y on 0<x<1, 0<y<1
Find P(X>Y).

Wribreeminsl 2021-05-05

A missile system's two electronic components must cooperate for the system as a whole to succeed. Let X and Y represent the combined life of the two components in hours. The joint density of X and Y is
$f (x, y) = y e^{- y (1 + x)}, x, y > 0$
Give the marginal density functions for both random variables. Are the independent?

Chaya Galloway 2021-05-02

Let X and Y be jointly continuous random variables wth joint PDF is given by:
$f_{X, Y} (x, y) = 2$ where $0 \leq y \leq x \leq 1$
Find $P (X \geq \frac{1}{2})$ .

Lennie Carroll 2021-05-01

Let X and Y be jointly continuous random variables wth joint PDF is given by:
$f_{X, Y} (x, y) = 2$ where $0 \leq y \leq x \leq 1$
Show the marginal PDF of X and Y.

Lewis Harvey 2021-03-09

A distribution of values is normal with a mean of 203.8 and a standard deviation of 50.2.
Find the probability that a randomly selected value is less than 113.4.
$P (X < 113.4) =$
Write your answers as numbers accurate to 4 decimal places.

When you have an equation that deals with random variables, always start with an explanation related to some random variable examples. It is what the majority of skilled engineers do because it is the only way how one can avoid mistakes when working with statistical data. Even though this part of statistics and probability may seem overly complex, approach every random variable equation through the lens of probability as you seek answers to your questions. Compare provided solutions, see similarities, and you will achieve the most efficient solutions.

Explore Random Variables: Examples & Equations