Probability - Continuous Random Variables Flashcards
what is the probability that X takes a certain value (P(X=x))
zero
what is (p.d.f)
the probability density function
what is the probability density function
a non-negative function such that P(x <= X < x + Δx) = f(x) Δx, the area of the region enclosed by y=f(x) between x and Δx
what is P(-∞ < X < + ∞)
1
what is P(X=a)
0
what is (c.d.f) F(X)
the cumulative distribution function
what is the cumulative distribution function
the probability that X takes a value smaller than X, F(X)=P(X<=x)
how do you find P(a<X<b) using F(X)
F(b)-F(a)
how do you find P(X>a) using F(X)
1-P(X<a) = 1-F(a)
what are the similarities and differences in E(X) and Var(X) between continuous and discrete r.v’s
the expectation for a continuous variable is the area under the curve xf(x) but variance is the same
what does X~U(a,b) mean
X is a random variable with uniform distribution on the interval [a,b]
what is the p.d.f if X~U(a,b)
f(x) = 1/b-a if a<=x<=b, otherwise 0
what is the c.d.f if X~U(a,b)
0 if x<a, x-a/b-a if a<=x<b and 1 if b<=x
what are E(X) and Var(X) if X~U(a,b)
E(X) = a+b/2
Var(X) = (b-a)^2/12
what does X~N(μ, σ^2) mean, what are μ and σ^2
X is a normally distributed random variable
μ is the expected value
σ^2 is the variance
what are some of the defining features of the normal p.d.f
bell shape, symmetric about μ, unimodal, standard deviation σ
how much of the area is within one and two σ from μ
68% and 95%
what does μ affect on the graph of f(x) for X~N(μ, σ^2)
it shifts the peak along the x axis
what is the effect of σ on the graph of f(x) for X~N(μ, σ^2)
the smaller the σ, the higher the peak
how do you calculate probabilities for normal r.v’s
work with the standard normal random variable
what is the standard normal random variable of the standard normal distribution Z~N(0,1)
Z=X-μ/σ
for X~N(μ, σ^2) how do you find P(a<=X<=b)
P(a-μ/σ<=Z<=b-μ/σ)
what is Φ(z)
the probability density function for the standard normal P(Z<=z)
what is Φ(z)=P(Z<=z)
the cumulative distribution function for the standard normal
what is equivalent to Φ(-z)
1-Φ(z) = P(Z<=-z)
how do you calculate P(a<Z<b)
Φ(b)-Φ(a)
If X ~ N(μx , σx^2) and Y ~ N(μy , σy^2) are independent, what is the distribution of aX+b
N(aμx+b, a^2σx^2)
If X ~ N(μx , σx^2) and Y ~ N(μy , σy^2) are independent, what is the distribution of X+Y
N(μx+μy, σx^2 +σy^2)
what is the sample mean
a new random variable when there are n independent r.v’s with the same mean and variance
what is the expectation of the sample mean
the mean shared by all the independent r.v’s
what is the variance of the sample mean
the variance shared by all the independent r.v’s squared divided by the number of independent r.v’s
if the r.v’s contributing to the sample mean are independent and normally distributed then what is the distribution of the sample mean
Xbar~N(μ,σ^2/n)
what happens to the variance of the sample mean when the sample is large
it tends to zero
the mean of a large random sample from a variable is likely to be a good estimate for the expectation of the variable
what is xp
the p-quantile of the distribution
what is p when xp is the upper quartile of X
0.75
what is p when xp is the lower quartile of X
0.25
what is p when xp is the median of X
0.5
if α is greater than 0 but less than 1, what is xα where P(X>=xα) = α
the critical value corresponding to α
what is another way of thinking of xα
the 1-α quantile
