Question

3. Let W = 2N, where N ~ Geometric(p). In class, we showed that when p = 1/2, E(W) = ?. For all the other values 0 < p < 1 p ? 1/2, find* E(W). When is it finite? *For #3, you may need the following fact from calculus: if 0 < r < 1, $\sum_{k=0}^{\infty} r^k = 1 + r + r^2 + ... = \frac{1}{1 - r}$. If r ? 1, then 1 + r + r2 + ... = ?.

          3. Let W = 2<sup>N</sup>, where N ~ Geometric(p). In class, we showed that when p = 1/2,
E(W) = ?. For all the other values 0 < p < 1 p ? 1/2, find* E(W). When is it
finite?
*For #3, you may need the following fact from calculus: if 0 < r < 1, $\sum_{k=0}^{\infty} r^k = 1 + r + r^2 + ... = \frac{1}{1 - r}$.
If r ? 1, then 1 + r + r<sup>2</sup> + ... = ?.

3. Let W = 2<sup>N</sup>, where N Geometric(p). In class, we showed that when p = 1/2,
E(W) = ?. For all the other values 0 < p < 1 p ? 1/2, find* E(W). When is it
finite?
*For #3, you may need the following fact from calculus: if 0 < r < 1, ∑k=0^∞ r^k = 1 + r + r^2 + ... = (1)/(1 - r).
If r ? 1, then 1 + r + r<sup>2</sup> + ... = ?.

Added by Joseph G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Ma. Theresa Alin

01/22/2022

Step 1

Therefore, in this case, E(W) = 2E(N) = 2/p. Show more…

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Let W = 2^(N), where N ~ Geometric(p). In class, we showed that when p = 1/2, E(W) = ∞. For all the other values E(W) >= 11 + r + r^2 + dots = ∞. If r >= 1, then 1 + r + r^2 + dots = ∞. Find E(W). When is it finite? (Continued) For #3, you may need the following fact from calculus: if 0 < r < 1, D = 0. If r > 1, then 1 + r + r^2 + dots = ∞. Let W = 2N where N ~ Geometric(p). In class, we showed that when p = 1/2, E(W) = ∞. For all the other values 0 1, then 1 + r + r^2 + ... = ∞.