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Chapter 1: Problem 1

Let \(X\) equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of \(X\) and compute the probability that \(X\) is equal to an odd number.

Short Answer

Expert verified
The probability that \(X\) is equal to an odd number is 0.5.

Step by step solution

01

Calculate the pmf using the binomial theorem

The pmf of a binomial distribution is given by \(f(k; n, p) = Pr(X = k) = C(n, k) (p^k) ((1 - p)^{(n - k)})\), where \(C(n, k)\) is the number of combinations of \(n\) items taken \(k\) at a time, \(p\) is the probability of success, and \(n - k\) is the number of failures. In this case, \(n = 4\) and \(p = 0.5\). Calculate \(f(k; 4, 0.5)\) for \(k = 0, 1, 2, 3, 4\).
02

Sum up the probabilities for an odd number of heads

The odd numbers within the range 0 to 4 are 1 and 3. Sum up \(f(1; 4, 0.5)\) and \(f(3; 4, 0.5)\) to get the probability of getting an odd number of heads.
03

Calculate the probabilities

Calculate the probabilities for each \(k\) and sum them up for odd k. \(f(1; 4, 0.5) = C(4, 1) * (0.5^1) * (0.5^3) = 0.25\) and \(f(3; 4, 0.5) = C(4, 3) * (0.5^3) * (0.5^1) = 0.25\). Add them together to get the probability of getting an odd number of heads: \(0.25 + 0.25 = 0.5\).

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Most popular questions from this chapter

Let \(X\) be a random variable of the continuous type that has pdf \(f(x)\). If \(m\) is the unique median of the distribution of \(X\) and \(b\) is a real constant, show that $$E(|X-b|)=E(|X-m|)+2 \int_{m}^{b}(b-x) f(x) d x$$ provided that the expectations exist. For what value of \(b\) is \(E(|X-b|)\) a minimum?

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Let \(f(x)=\frac{1}{3},-1
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