Probability chapter 3

Question 1

What is a continuous random variable X(s)?

Answer 1

A continuous random variable X(s) is a sample random variable whose sample space S has an uncountable number of outcomes. An uncountable sample space is when the outcomes cannot be listed in order of size.

Question 2

How do you convert a PDF to a CDF and vice versa?

Answer 2

Integrate the PDF f(x) to get the CDF f(x)

Differentiate the CDF f(x) to get the PDF f(x)

Question 3

What is a CDF defined as?

Answer 3

F(r)=P(x<=r)= integral (upperlimits=r)(lowerlimits=-infinity) f(x) dx

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Question 4

How do you calculate the probability of X being within a given interval for continuous random variables?

Answer 4

See notes

Question 5

What is the expectation of a continuous random variable x given by?

Answer 5

The expectation of a continuous random variable is given by

E(X)=integral(upperlimit=infinity)(lowerlimit=-infinity) xf(x) dx= μx

Question 6

What is the normal distribution called?

Answer 6

The normal distribution is often called the Gaussian distribution. The PDF is given by

f(x)=1/(2π σ²)^1/2 exp(-(x-μ)/2 σ²)^2, -∞

Question 7

What are the properties of the normal distribution?

Answer 7

• E(X)=μ
• Var(X)= σ²
• sd(x)= σ

A normal random variable would be denoted by X~N(μ, σ²)

Question 8

What do we do each time we approximate a discrete distribution with a continuous one?

Answer 8

We make the following corrections:

• P(X>r) is replaced with P(X>r+0.5)
• P(X ≥r) is replaced with P(X ≥r-0.5)
• P(X

Question 9

How do you calculate the mean & the variance for a uniform distribution?

Answer 9

• E(X)=(a+b)/2

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