STAB22H3 Chapter Notes - Chapter 14: Bernoulli Distribution, Random Variable, Squared Deviations From The Mean

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Random variable: is a variable that takes a result from an experiment and gives it a numerical value. For example, when a coin is tossed, the random variable, x, can be defined as: Bernoulli random variable: a variable that takes the value of 1 with a probability of p and the value of 0 which has a probability of q = 1 p. Probability distribution (probability model): is list of values and probabilities that can be obtained from an event. For example, the probability distribution of the number of h (heads) when tossing. To find the mean (expected value) of a random variable, multiply each x value by it"s probability and then add the products together. The mean is represented by e(x) or . Since the median is the midpoint of a specific distribution, there will be an equal probability of falling above or below it.

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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

1. In its standardized form, the normal distribution

(a) has a mean of 0 and a standard deviation of 1.

(b) has a mean of 1 and a variance of 0.

(d) cannot be used to approximate discrete probability distributions.

2. For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3770.

The value of Z is

(a) 0.18

(b) 0.81

(d) 1.47

3. For some value of z, the probability that a standard normal variable is below z is 0.2090. The value of Z is

(a) -0.81

(b) -0.31

(d) 1.96

4. For some positive value of X, the probability that a standard normal variable is between 0 and +2x is 0.1255.

The value of X is

(a) 0.99

(b) 0.40

(d) 0.16

5. If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 35. minutes and a standard deviation of minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes.

(a) 0.3551

(b) 0.3085

(d) 0.1915

corallouse410

Consider the following sample of observations on coating thickness for low-viscosity paint. ("Achieving a Target Value for a Manufacturing Process: A Case Study," J. of Quality Technology, 1992:22-26):

0.83	0.88	0.88	1.04	1.09	1.12	1.29	1.31
1.48	1.49	1.59	1.62	1.65	1.71	1.76	1.83

Assume that the distribution of coating thickness is normal (a normal probability plot strongly supports this assumption).

(a) Calculate a point estimate of the mean value of coating thickness, and state which estimator you used.

(b) Calculate a point estimate of the median of the coating thickness distribution, and state which estimator you used.

(c) Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state which estimator you used. [Hint: Express what you are trying to estimate in terms of m and s]

(d) Estimate P(X<1.5), i.e., the proportion of all thickness values less than 1.5 [Hint: if you know the values of and , you could calculate this probability. These values are not available , but they can be estimated.]

(e) What is the estimated standard error of the estimator that you used in part (b)?

skybat612

STAB22H3 Chapter Notes - Chapter 14: Bernoulli Distribution, Random Variable, Squared Deviations From The Mean

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