13 Continuous Random Variables Flashcards
What is a continuous random variable?
It can take any value in an interval. Some variables may be treated as continuous even though they are discrete e.g a family’s annual income
What is the PDF denoted as
f(x)
What is the PDF
Probability Density function. It is the probability per unit value of the random variable, it can exceed 1
What is true if f(x)=0
x is outside the range that X is defined
How do we calculate the probability of a continuous random variable?
We integrate its PDF over the appropriate range of values
How is the CDF denoted?
F(x)
What is the CDF
The cumulative distribution function. It expresses the probability that X does not exceed the value x
Rules for PDF
f(x) must be positive
Integral of f(x) must =1
Rules for CDF
0<=F(x)<=1
What is the joint probability density function?
The probability density function for two continuous variables.
How can multiple integrals be evaluated?
By integrating with respect to one variable and treating the other as fixed
What is the marginal density function?
The marginal density function of X assigns probabilities to a range of values x, irrespective of the values Y can take
How can the joint PDF tell us when random variables are statistically independent
X and Y are independent if
f(x,y)= f(x)f(y)
How is the expected value worked out
The integral of xf(x)
How can the expected value tell us whether x and y are statistically independent
If E(XY)=E(X)E(Y) Then X and Y are independent
How do you work out the variance of a continuous random variable X
Var(X)=E(X^2)-E(X)^2
How do you work out the covariance of two continuous random variables X and Y
Cov(X,Y)= E(XY)- E(X)(E(Y)
How can the covariance tell us if x and y are independent
They are independent if cov(x,y)=0
Uniform distribution
Where all possible outcomes have equal probability. All values have equal density
What is the PDF of a uniform distribution?
f(x)=1/(b-a)
Where a<=x<=b
What is the CDF of a uniform distribution for z is a member of (a,b)
F(z)= P(x<=z)= (z-a)/(b-a)
What is the expected value of a uniform distribution?
E(X)=(b+a)/2
What is the variance of a uniform distribution?
Var(X)= E(X^2)-E(X)^2 =(b-a)^2/12
What can we say about normal distribution?
It is bell shaped with equal mean, median and mode. There is an infinite number of distributions
How is a normally distributed variable denoted?
X~N(u,ó^2)
What is the expected value for a normally distributed variable?
The mean
What is a standard normal distribution?
A normal distribution with mean=0 and variance=1
How do we standardise any normal random variable?
Z=(X-u)/ó
Subtract the mean and divide by its standard deviation
When can binomial distribution be approximated to normal distribution
When the number of tests (n) is large
