excans:5974505031: Difference between revisions
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(Created page with "'''Solution: D''' We let <math>X_i = \pm 1 </math> with probability that <math>X_i = 1 </math> equals 1/2. Then we want to approximate the probability that <math>\sum_{i=1}^{100}X_i \in [-10,10] </math>. The expected value of <math>X_i </math> is 0 and its variance is 1. By the central limit theorem, the sum <math>\sum_{i=1}^{100}X_i </math> is approximately normally distributed with mean 0 and variance 100. Hence we have <math display = "block">\sum_{i=1}^{100}X_i \i...") |
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'''Solution: D''' | '''Solution: D''' | ||
We let <math>X_i = \pm 1 </math> with probability that <math>X_i = 1 </math> equals 1/2. Then we want to approximate the probability that <math>\sum_{i=1}^{100}X_i \ | We let <math>X_i = \pm 1 </math> with probability that <math>X_i = 1 </math> equals 1/2. Then we want to approximate the probability that <math>\sum_{i=1}^{100}X_i \notin [-10,10] </math>. The expected value of <math>X_i </math> is 0 and its variance is 1. By the central limit theorem, the sum <math>\sum_{i=1}^{100}X_i </math> is approximately normally distributed with mean 0 and variance 100. Hence we have | ||
<math display = "block">\sum_{i=1}^{100}X_i \ | <math display = "block">\sum_{i=1}^{100}X_i \notin [-10,10] \approx P(Z \notin [-1,1])</math> | ||
where <math>Z</math> is a standard normal. This is approximately equal to 0. | where <math>Z</math> is a standard normal. This is approximately equal to 0.3173. | ||
Latest revision as of 17:41, 28 June 2024
Solution: D
We let [math]X_i = \pm 1 [/math] with probability that [math]X_i = 1 [/math] equals 1/2. Then we want to approximate the probability that [math]\sum_{i=1}^{100}X_i \notin [-10,10] [/math]. The expected value of [math]X_i [/math] is 0 and its variance is 1. By the central limit theorem, the sum [math]\sum_{i=1}^{100}X_i [/math] is approximately normally distributed with mean 0 and variance 100. Hence we have
[[math]]\sum_{i=1}^{100}X_i \notin [-10,10] \approx P(Z \notin [-1,1])[[/math]]
where [math]Z[/math] is a standard normal. This is approximately equal to 0.3173.