Concept explainers
This exercise demonstrates that, in general, the results provided by Tchebysheff’s theorem cannot be improved upon. Let Y be a random variable such that
a Show that E(Y) = 0 and V(Y) = 1/9.
b Use the
*c In part (b) we guaranteed E(Y) = 0 by placing all probability mass on the values –1, 0, and 1, with p(–1) = p(1). The variance was controlled by the probabilities assigned to p(–1) and p(1). Using this same basic idea, construct a probability distribution for a random variable X that will yield
*d If any k > 1 is specified, how can a random variable W be constructed so that
a
Prove that
Explanation of Solution
Calculation:
The expected value of a random variable is the sum of the product of each observation with corresponding probability.
Hence, the expected value is obtained as follows:
Similarly the variance of a random variable is the expected value of the squared deviation from the mean.
Hence, it is proved that
b
Find the value of
Compare the exact probability with the upper bound of the Tchebysheff’s theorem.
Answer to Problem 169E
The value of
Explanation of Solution
Calculation:
Tchebysheff’s Theorem:
Let Y be a random variable with mean
Thus, using the Tchebysheff’s theorem it can be obtained that,
Similarly, using the given information the required probability is obtained as follows:
Hence, the value of
Thus, it can be said the exact probability is same with the upper bound of the Tchebysheff’s theorem when
c.
Generate a probability distribution for a random variable X that will yield
Answer to Problem 169E
The probability distribution of another sufficient random variable is,
Explanation of Solution
Calculation:
Consider the random variable X which is symmetric around mean 0.
The probability distribution of X is,
It is needed to find the value of p.
The mean of X is,
The variance of X is,
Thus, using Tchebysheff’s theorem it can be said that,
Now, it is obvious that
Hence,
Thus, the probability distribution of another sufficient random variable is,
d.
Explain the procedure of obtaining a random variable W such that
Explanation of Solution
Calculation:
Consider the random variable W which is symmetric around mean 0.
The probability distribution of W is,
It is needed to find the value of p.
The mean of W is,
The variance of W is,
Thus, using Tchebysheff’s theorem it can be said that,
It is clear that W can take only one positive value. Hence, if
Hence,
Thus, the probability distribution of another sufficient random variable is,
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Chapter 3 Solutions
Mathematical Statistics with Applications
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