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Statistical Sciences 2035 Chapter Notes - Chapter 5: Standard Deviation, Random Variable, Probability Distribution

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School
Western University
Department
Statistical Sciences
Course
Statistical Sciences 2035
Professor
Histro Sendov

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Document Summary

Continuous random variable: may assume any numerical value in one or more intervals. The total area under the curve of f(x) is equal to 1. P(x = a) = p(x = b) since they are of a line is 0. A population that has continuous distribution is a very large population in which only a few members are equal to a. Therefore, proportion of the population that is equal to a is 0. If a ~ b, then the area under f(x) between a and b is ~ f(a) x (b a) Mean = c + d / 2 = median standard deviation . Chebyshev"s theorem: intervals that contain a specified percentage of the measurements from a population are called tolerance intervals. The empirical rule holds for distributions that are normal or close to normal. In general, we have: let and be the population"s mean and standard deviation.

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Related Questions

For a continuous random variable x, the probability density function f(x) represents
 
a. the probability at a given value of x
 
b. the area under the curve at x
 
c. the probability at a given value of x and the area under the curve at x are correct answers.
 
d. the height of the function at x

Q5. Let (Ω, F , P) be a probability space. Let λ > 0 be some constant.

  1. (a)  Give the definition of a random variable X on R ∪ {−∞, +∞}.

  2. (b)  Define the distribution function FX for the RV X.

[6 marks]

[1 marks]

[1 marks] (c) Suppose that Y is a RV on R ∪ {−∞, +∞} with distribution function given by

􏰄3 −1e−λy (y≥0)

[6 marks]

page1image62001664 page1image62003968

FY (y) = 4 2 1

(y < 0).
For any a < b, determine P(Y > a), and P(Y ∈ (a, b]). [3 marks]

  1. (d)  By noting {Y < b} = ∪n∈N{Y ≤ b − 1 } and using Continuity of Probability, n

    find P(Y < b). Similarly, find P(Y < ∞) and P(Y = −∞). [4 marks]

  2. (e)  Is Y either a discrete RV, or a continuous RV, or a mixture of both? Explain

    your answer. [1 marks]

The joint probability density function of X and Y is given by

f(x,y)={e-(x+y)   for X>0, y>0      and    0   elsewhere

A. Find the marginal density of X

B. Find the marginal density of Y

C. Find the Conditional density of X given Y

D. Are random variables X and Y independent? State the reason of your answer.

E. Find P(X<.5, y<.5)

F. Find P(X=.5, y<.5)