Random Variable Problems & Solutions

For continuous random variables X and Y with joint probability density function
$f (x, y) = {\begin{cases} x e^{- (x + x y)} & x > 0 a n d y > 0 \\ 0 & o t h e r w i s e \end{cases}$
Find P(X>1 and Y>1).

Trent Carpenter 2021-11-03

The joint density of the random variables X and Y is given by
$f (x, y) = {\begin{cases} 8 x y & 0 \leq x \leq 1, 0 \leq y \leq x \\ 0 & o t h e r w i s e \end{cases}$
Find the conditional density of X
Find the conditional density of Y

sanuluy 2021-11-01

Assume that X and Y are jointly continuous random variables with joint probability density function given by
$f (x, y) = b e g \in {c a s e s} \frac{1}{36} (3 x - x y + 4 y) if 0 < x < 2 and 1 < y < 3 ∖ 0 o t h r e w i s e e n d {c a s e s}$
Find the marginal density functions for X and Y .

Suman Cole 2021-11-01

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 X.

generals336 2021-10-28

Suppose that the random variables X and Y have joint p.d.f.
$f (x, y) = {\begin{cases} k x (x - y), 0 < x < 2, - x < y < x \\ 0, e l s e w h e r e \end{cases}$
Find the marginal p.d.f. of the two random variables.

nitraiddQ 2021-10-28

Random variable X and Y have the joint PDF
$f_{X Y} (x, y) = {\begin{cases} c, x \geq 0, y \geq 0, (x^{2} + y) \leq 1, \\ 0, o t h e r w i s e \end{cases}$
Find the marginal PDF $f_{Y} (y)$ .

banganX 2021-10-28

Suppose that X and Y are continuous random variables with joint pdf $f (x, y) = e^{- (x + y)} 0 < x < \infty and 0 < y < \infty$ and zero otherwise.
Find P(X+Y>3)

hexacordoK 2021-10-27

Assume that X and Y are jointly continuous random variables with joint probability density function given by
$f (x, y) = b e g \in {c a s e s} \frac{1}{36} (3 x - x y + 4 y) if 0 < x < 2 and 1 < y < 3 ∖ 0 o t h r e w i s e e n d {c a s e s}$
Find Cov(X,Y).

abondantQ 2021-10-26

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

jernplate8 2021-10-26

Consider two continuous random variables X and Y with joint density function
$f (x, y) = b e g \in {c a s e s} x + y o \leq x \leq 1, 0 \leq y \leq 1 ∖ 0 o t h e r w i s e e n d {c a s e s}$
P(X>0.8, Y>0.8) is?

BenoguigoliB 2021-10-25

Suppose that X and Y are continuous random variables with joint pdf $f (x, y) = e^{- (x + y)} 0 < x < \infty and 0 < y < \infty$ and zero otherwise.
Find P(X>Y)

Tahmid Knox 2021-10-25

Random variable X and Y have the joint PDF
$f_{X Y} (x, y) = {\begin{cases} c, x \geq 0, y \geq 0, (x^{2} + y) \leq 1, \\ 0, o t h e r w i s e \end{cases}$
Find the marginal PDF $f_{X} (x)$ .

Dolly Robinson 2021-10-23

Let $X_{1}, X_{2}, \dots, X_{n}$ be n independent random variables each with mean 100 and standard deviation 30. Let X be the sum of these random variables.
Find n such that $P r (X > 2000) \geq 0.95$ .

Wierzycaz 2021-10-23

Random variable X and Y have the joint PDF
$f_{X Y} (x, y) = {\begin{cases} c, x \geq 0, y \geq 0, (x^{2} + y) \leq 1, \\ 0, o t h e r w i s e \end{cases}$
Find the constant c.

Daniaal Sanchez 2021-10-22

Let $X_{1}, X_{2}, \dots, X_{6}$ 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 $X_{6}$ .

Albarellak 2021-10-21

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.

Falak Kinney 2021-10-20

Random variables $X_{1}, X_{2}, \dots, X_{n}$ are independent and identically distributed; 0 is a parameter of their distribution.
If $X_{1}, X_{2}, \dots, X_{n}$ are Normally distributed with unknown mean 0 and standard deviation 1, then $\overset{―}{X} \sim N (\frac{0, 1}{n})$ . Use this result to obtain a pivotal function of X and 0.

amanf 2021-10-19

Random variables $X_{1}, X_{2}, \dots, X_{n}$ are independent and identically distributed; 0 is a parameter of their distribution.
If $q (X, 0) \sim N (0, 1)$ is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

pedzenekO 2021-10-19

Suppose that $X_{1}, X_{2} and X_{3}$ are three independent random variables with the same distribution as X.
What is the ecpected value of the sum $X_{1} + X_{2} + X_{3}$ ? The product $X_{1} X_{2} X_{3}$ ?
Suppose a discrete random variable X assumes the value $\frac{3}{2}$ with probability 0.5 and assumes the value $\frac{1}{2}$ with probability 0.5.

Cabiolab 2021-10-19

For continuous random variables X and Y with joint probability density function
$f (x, y) = {\begin{cases} x e^{- (x + x y)} & x > 0 a n d y > 0 \\ 0 & o t h e r w i s e \end{cases}$
Are X and Y independent? Explain.

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