Explore Random Variables: Examples & Equations

An initial statistics course's final exam grades have a mean of 73 and a standard deviation of 8, and they are normally distributed. A score of 84 represents how many standard deviations from the mean?

1. Given a random sample size of n = 900 from a bi- nomial probability distribution with P = 0.10 do the following:

   1. Find the probability that the number of successes is greater than 110.

   2. Find the probability that the number of successes is fewer than 53.

   3. Find the probability that the number of successes is between 55 and 120.

   4. With probability 0.10, the number of successes is fewer than how many?

   5. Withprobability0.08,thenumberofsuccessesis greater than how many?

Given a standard normal distribution, find the area under the curve that lies 
(a) to the left of ${\displaystyle z=-1.39}$; 
(b) to the right of ${\displaystyle z=1.96}$; 
(c) between ${\displaystyle z=-2.16\text{and}z=-0.65}$; 
(d) to the left of ${\displaystyle z=1.43}$; 
(e) to the right of ${\displaystyle z=-0.89}$; 
(f) between ${\displaystyle z=-0.48\text{and}z=1.74}$.

Let's say Z=X+Y, where X and Y are independent uniform random variables with range [0,1]. Then the PDF is 

$f\left(z\right)=\{\begin{array}{ll}z& for 0<z<1\\ 2-z& x>for 1\le z<2\\ 0& otherwise.\end{array}$

How was this PDF obtained?

Given that z is a standard normal random variable, compute the following probabilities.
a. ${\displaystyle P(z\le -1.0)}$
b. ${\displaystyle P(z\ge -1)}$
c. ${\displaystyle P(z\ge -1.5)}$
d. ${\displaystyle P(-2.5\le z)}$
e. ${\displaystyle P(-3<z\le 0)}$