Midterm to Week 7 Flashcards
Distinguish between discrete and continuous random variables.
Discrete: Assume only a countable number of values.
Continuous: Random variable can be any point contained in an interval.
In a continuous system, what is the probability that the random variable X equals exactly x for any x. Why?
0
A continuous system works off ranges.
What is the name of the probability function for continuous system?
Probability Density Function.
What condition is applied to function f(x) for a proper probability distribution?
f(x) >= 0 For all x
The sum of all the probabilities is 1.
What is the expected value of E(X)?
+∞∫-∞ x * f(x) dx
What is the expected value of E(g(x))?
+∞∫-∞ g(x) f(x) dx
What is Var(X)?
∫x^2 f(x) dx - (∫ x f(x) dx )^2
Formula for uniform probability density function.
f(x) = { 1 / (b - a)
if a <= x <= b
{ 0
otherwise
Formula for uniform cumulative distribution function.
F(x) = x∫a f(u) du = x∫a 1 / (b - a) du =
0 if x < a
(x - a) /( b - a) if a<= x <= b
1 if x > b
What is the mean and variance of the uniform distribution function?
Mean: a + b / 2
Variance = (b - a)^2 / 12
P(c < x < d) = F(d) - F(c) = (d - c) / (b - a) where a <= c < d <= b
Formula for exponential probability density function. Expected value? Variance? What condition is placed on λ?
f(x) = { λe^-λx if x >= 0
{ 0 otherwise
E(X) = 1 / λ
Variance = 1 / (λ)^2
λ > 0
Formula for exponential cumulative distribution
P(X > x) = e^(-λx)
Memoryless Property Formula:
P(X > a + b|X > a) = P(X > b)
Describe the relationship between the Poisson and exponential distributions.
The time between consecutive events of a Poisson(λ) process follows an exponential distribution with the same rate λ.
If a value is in between two values in the statistical tables which value should be taken higher or lower value.
Always round up from 0.5
