Explore Random Variables: Examples & Equations

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of
${\displaystyle 1-{(1-p)}^2}$ if it does rain
${\displaystyle 1-p^2}$ if it does not rain
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability ${\displaystyle p\cdot }$, what value of p should he or she assert so as to maximize the expected score?

The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320,000, what fraction earn (a) below 250,000; (b) between 260,00 and 300,000?

The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is the probability that there will be (a) more than 2 such accidents in the next month? (b) more than 4 such accidents in the next 2 months? (c) more than 5 such accidents in the next 3 more than 5 such accidents in the next 3 months? Explain your reasoning!

The annual rainfall (in inches) in a certain region is normally distributed with ${\displaystyle \mu =40}$ and ${\displaystyle \sigma =4}$
What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?

Suppose that X is a normal random variable with mean 5. If P{X>9}=.2, approximately what is Var(X)?

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Are X and Y independent?

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X<20]

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find Var(Y).

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find E[Y].

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find Var(X).

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find E[X].

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X>16]

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X<8]

Let X be a normal random variable with mean 12 and variance 4. Find the value of c such that P[X>c]=.10.

Random variables X and Y have joint PDF
$f_{X,Y}(x,y)=\{\begin{array}{l}12e^{-(3x+4y)},x\ge 0,y\ge 0\\ 0,otherwise\end{array}$
Find ${\displaystyle P[X+Y\le 1]}$

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X>5]

Let ${\displaystyle X_1\dots .,X_n\text{and}{Y}_1,\dots ,Y_m}$ be two sets of random variables. Let ${\displaystyle a_i,b_j}$ be arbitrary constant.
Show that
${\displaystyle Cov(\sum _{i=1}^n{a}_i{X}_i,\sum _{j=1}^m{b}_j{Y}_j)=\sum _{i=1}^n\sum _{j=1}^m{a}_i{b}_{j}Cov(X_i,Y_j)}$

Random variables X and Y have joint PDF
$f_{X,Y}(x,y)=\{\begin{array}{l}12e^{-(3x+4y)},x\ge 0,y\ge 0\\ 0,otherwise\end{array}$
Find ${\displaystyle P[min(X,Y)\ge 2]}$

Compute the distribution of X+Y in the following cases:
X and Y are independent normal random variables with respective parameters ${\displaystyle ({\mu }_1,{\sigma }_1^2)\text{and}({\mu }_2,{\sigma }_2^2)}$.