13 Continuous Random Variables

Question 1

What is a continuous random variable?

Answer 1

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

Question 2

What is the PDF denoted as

Answer 2

f(x)

Question 3

What is the PDF

Answer 3

Probability Density function. It is the probability per unit value of the random variable, it can exceed 1

Question 4

What is true if f(x)=0

Answer 4

x is outside the range that X is defined

Question 5

How do we calculate the probability of a continuous random variable?

Answer 5

We integrate its PDF over the appropriate range of values

Question 6

How is the CDF denoted?

Answer 6

F(x)

Question 7

What is the CDF

Answer 7

The cumulative distribution function. It expresses the probability that X does not exceed the value x

Question 8

Rules for PDF

Answer 8

f(x) must be positive

Integral of f(x) must =1

Question 9

Rules for CDF

Answer 9

0<=F(x)<=1

Question 10

What is the joint probability density function?

Answer 10

The probability density function for two continuous variables.

Question 11

How can multiple integrals be evaluated?

Answer 11

By integrating with respect to one variable and treating the other as fixed

Question 12

What is the marginal density function?

Answer 12

The marginal density function of X assigns probabilities to a range of values x, irrespective of the values Y can take

Question 13

How can the joint PDF tell us when random variables are statistically independent

Answer 13

X and Y are independent if

f(x,y)= f(x)f(y)

Question 14

How is the expected value worked out

Answer 14

The integral of xf(x)

Question 15

How can the expected value tell us whether x and y are statistically independent

Answer 15

If E(XY)=E(X)E(Y)
Then  X and Y are independent

Question 16

How do you work out the variance of a continuous random variable X

Answer 16

Var(X)=E(X^2)-E(X)^2

Question 17

How do you work out the covariance of two continuous random variables X and Y

Answer 17

Cov(X,Y)= E(XY)- E(X)(E(Y)

Question 18

How can the covariance tell us if x and y are independent

Answer 18

They are independent if cov(x,y)=0

Question 19

Uniform distribution

Answer 19

Where all possible outcomes have equal probability. All values have equal density

Question 20

What is the PDF of a uniform distribution?

Answer 20

f(x)=1/(b-a)

Where a<=x<=b

Question 21

What is the CDF of a uniform distribution for z is a member of (a,b)

Answer 21

F(z)= P(x<=z)= (z-a)/(b-a)

Question 22

What is the expected value of a uniform distribution?

Answer 22

E(X)=(b+a)/2

Question 23

What is the variance of a uniform distribution?

Answer 23

Var(X)= E(X^2)-E(X)^2 =(b-a)^2/12

Question 24

What can we say about normal distribution?

Answer 24

It is bell shaped with equal mean, median and mode. There is an infinite number of distributions

Question 25

How is a normally distributed variable denoted?

Answer 25

X~N(u,ó^2)

Question 26

What is the expected value for a normally distributed variable?

Answer 26

The mean

Question 27

What is a standard normal distribution?

Answer 27

A normal distribution with mean=0 and variance=1

Question 28

How do we standardise any normal random variable?

Answer 28

Z=(X-u)/ó

Subtract the mean and divide by its standard deviation

Question 29

When can binomial distribution be approximated to normal distribution

Answer 29

When the number of tests (n) is large