Chapter 4: Continuous random variables and probability distributions
standard normal distribution
- the Normal distribution N(0, 1) with mean 0, standard dev. 1 - if var. x has any Normal distribution N(µ, σ) with mean µ and standard dev. σ, then the standardized variable z has the standard normal distribution
uniform distribution sample problem
A = 3, B = 0
percentile
A point on a ranking scale of 0 to 100. The 50th percentile is the midpoint; half the people in the population being studied rank higher and half rank lower.
what is a pdf?
In other words, a probability distribution / pdf for a continuous r.v. is a mathematical formula that defines the possible values of the r.v. and the probabilities corresponding to intervals of values.
normal distribution
It is often used in the natural and social sciences to approximate the distributions of real-valued r.v.s. random variable can range from -∞ to ∞
Which of the following statements best describes P(X < c) as it relates to the probability distribution function (pdf), f(x)?
It is the area under f(x) to the left of c.
pdf "not a probability" misconception
P(x) IS a probability it's just the FUNCTION f(x) that we are integrating that is NOT a probability
difference between pdf and cdf
PDF → probability density function CDF → Cumulative density function Probability looks at probability at one point. Cumulative is the total probability of anything below it. As you can see in the diagram below, the cumulative is much greater than the just probability because it is the sum of many, and not just of one probabilities.
Phi(greek letter that looks like and I with a circle in the middle)
The cumulative probability, P(Z ≤z) is equivalent to the area under the curve to the left of z and is denoted by the symbol in the image aka phi.
75th percentile question
answer is 0.67
flaw between 0.40 and 0.45 problem
can either solve using pdf or cdf
Uniform Distribution
can use this instead of integrating from negative inifinity to infinity if the situation is right
F(x) is...
cdf
difference between pdf and cdf formula wise
cdf is a pdf except the lower bound is -infinity aka every number up to the upper bound pdf is f(x) itself NOT THE INTEGRAL
percentile part 2
everything up to that point
(image) flaw problem cdf
everything up to the random variable note: instead of negative infinity, use 0 for lower bound
derivatives and integrals tip
f(x) is the function aka base the integral is ∫f(x) the derivative is f'(x) but remember, f(x) is the BASE
note about F(x)
if you see F(x) that means that f(x) is already integrated.
cdf is the _____ of pdf
integral
a pdf is not a probability. that means that
it can go above one but to be legitimate it has to be equal to 1
for a cdf, F(x) can never go above one because...
it is a probability
E(x) aka mean
just integrate from negative infinity to infinity note: in our case it's just from zero to one
cdf accounts for areas outside of graph?
maybe
η
mean
cdf
note the negative infinity
cumalitive probabilty of standard normal distribution
note: F is the cdf(cumulative shit)
(image)flaw problem part 2
note: can also do 1 - area in between
(image)flaw problem
note: can also do 1 - unshaded region
P(Z > z₀) = 0.95
note: the notation means the area greater than(to the right) of z₀ is 0.95
sample problem: find the 90th percentile of X
note: η(0.90) is a point within the curve, just plug it into the function i.e (η(0.90)²)
f(x) is...
important note about pdf
pdf is f(x) itself NOT THE INTEGRAL
for cdf
plug little x into the funciton
remember that pdf is not...
pmf
probability density function
same as pmf except this is for continuous variables
What does legitimate pdf mean?
set it equal to 1
σ
standar deviation
P(-1 < x < 1)
take everything up to x = 1 and subtract everything up to x = -1 to get the stuff in between
pdf can be formulated by...
taking the derivative of cdf F(x) = left tail of curve aka the cdf
E[X] mean
the expected(estimated) value
The more specific the measurement...
the smoother the curve
standardizing
transforming a distribution to a standard normal distribution
what does variance = infinity mean
we can't measure the spread
if x < 0.30 can x be a negative number?
yes but we've already accounted for that situation as seen in the image
90th percentile in height means...
you're taller than 90% of the population
