Download Homework 3

MIME 5100
Due 2/19/2016
Homework 3
2/18/2016
1. Read section C2 (pp. 571-579) of appendix C of the textbook.
2. X is a discrete random variable with the following probability mass function:
Value
1
2
3
4
5
Probability
0.1
0.3
0.3
0.2
0.1
a) Check if this is a valid probability mass function. Hint: Probabilities of outcomes
range from 0 to 1, and the sum of the probabilities of all possible outcomes is always 1.
b) Plot the probability mass function fX(x).
c) Calculate and plot the cumulative distribution function FX(x).
d) Find (1.4 ≤ 𝑋 ≤ 4.2) .
e) Find P(1.4<X<4.2).
f) Find the expected (mean) value of X.
g) Find the variance of X, Var(X).
h) Find the standard deviation of X, 𝜎𝑋 .
3. X is a uniformly distributed random variable from 10 to 20. That means the probability
density function of X is constant from 10 to 20 and zero elsewhere.
a) Find the mean values of X.
b) Find the variance of X.
c) Find the probability that 𝑋 < 12.
d) Find the probability that X >12.
e) Find the probability that X<8 or X >22.
4. X is an exponentially distributed random variable with a mean value of 5. The
cumulative distribution of this variable is:
𝑥
𝐹𝑋 (𝑥) = 1 − 𝑒 5 𝑓𝑜𝑟 𝑥 ≥ 0 and 0 elsewhere.
a) Find the probability 𝑃(𝑋 > 5).
b) Find 𝑃(1.4 ≤ 𝑋 ≤ 4.2).
c) Find 𝑃(1.4 < 𝑋 < 4.2).
Reference: Kelton, W. D, et al., Simulation with Arena, 4th edition, McGraw Hill, 2007.