Suppose you have a coin that lands on Heads with probability 0.7. Define the ran

Question

Suppose you have a coin that lands on Heads with probability 0.7. Define the random variable X as the sum of three coin flips, (where heads count 1 and tails count 0 - e.g., if three heads then X= 3: if a head and two tails then X = 1: if three tails the X = 0: ...). Compute the expected value of X. b) What is the probability distribution function of X (i.e. state all of the possible values for X and the probability of X equaling each of those values) c) What is the variance of X? d) How many times do you need to Hip a fair coin (one where the probability of a head is 0.5) so that the probability of getting ALL heads is less than 0.001 (i.e. one out of one thousand).

Explanation / Answer

A) expected value of X = n * P = 3 * 0.7 = 2.1

B) P(X = 0) = 3C0 * (0.7)0 * (0.3)3 = 0.027

P(X = 1) = 3C1 * (0.7)1 * (0.3)2 = 0.189

P(X = 2) = 3C2 * (0.7)2 * (0.3)1 = 0.441

P(X = 3) = 3C3 * (0.7)3 * (0.3)0 = 0.343

C) variance of X = n * P * (1 - P) = 3 * 0.7 * 0.3 = 0.63

D) let no of flip = n

P = 0.5

0.5n < 0.001

Or, ln(0.5n) > ln(0.001)

Or, n * ln(0.5) > ln(0.001)

Or, n > ln(0.001) / ln(0.5)

Or, n > 9.96

n = 10

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