STAT 2040 Lecture Notes - Lecture 6: Statistical Inference, Probability Distribution, Probability Density Function

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ochreferret449

1 Apr 2015

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Continuous random variables can take on an infinite number of possible values, will not have a countable number of values. P ( x = a ) = 0 for all: for any constant a, p(x a) = p(x > a) and p(x a) = p(x < a) a . Standard uniform distribution the most common continuous uniform distribution (c = 0; d = The greater the standard deviation, the more spread out the distribution and the lower the peak. Find z such that area to the left is 0. 8212 p(zqnorm(. 8212) Find 95th percentile z such that p(zqnorm(0. 95) *if a random variable x has the normal distribution, then it can take on any infinite valu. Linear transformation: the x2 distribution is related to the standard normal distribution. The mean: for k 2, the mode of the x2 distribution occurs at k- 2. For k 2 the mode occurs at 0.

Related Questions

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

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

FY (y) = 4 2 1

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

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

your answer. [1 marks]

amaranthwalrus338

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

Y=0 Y=1

X=0
X=1

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?

brownhyena716

Consider 3 trials, each having the same probability of success. Let X denote the total number of successes in these trials. If E[X]=1.8, what is (a) the largest possible value of P{X=3}? (b) the smallest possible value of P{X=3}? In both cases, construct a probability scenario that results in P{X=3} having the stated value. Hint: For part (b), you might start by letting U be a uniform random variable on (0, 1) and then defining the trials in terms of the value of U.

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STAT 2040 Lecture Notes - Lecture 6: Statistical Inference, Probability Distribution, Probability Density Function

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Introductory Statistics

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