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
Formula for table use. E.g. to find P(Z < 2) for mean 4 and variance 16.
P(x - u / sigma < 2 - 4 / root(16)) = P(Z < -0.5) for mean 0 and variance 1.
What is expected value of an exponential random variable?
1 / rate
