Flip a biased coin 90 times. On each flip, P[H] = 0.75. (a) Let X_i denote the n
ID: 3220848 • Letter: F
Question
Flip a biased coin 90 times. On each flip, P[H] = 0.75. (a) Let X_i denote the number of heads that occur on flip i. Then X_78 is a ()random variable. That is, identify the type of random variable and its parameter(s). If there is more than one parameter, then separate by a comma. (b) Define Y = X_1 + X_2 +...+X_90. Then Y is a () random variable. That is, identify the type of random variable and its parameter(s). If there is more than one parameter, then separate by a comma. (c) E[Y] = (d) Var[Y] =Explanation / Answer
a) X78 is a Binomial Random variable. n = 78 , p = 0.75
n indicates number of flips. p is the probability of success that is occurence of head on each trial or each flip.
Sometimes we also say it is discreet binomial random variable. But, in this context it is obvious.
b) Y is Binomial , n =4095 and p = 0.75
Y is a sum of binomial variables with equal probability parameter
Therefore Y itself would be binomially distributed.
Y is binomial with n = 1+2+3.. + 90 = 4095 AND p = 0.75
c) Expectation of Y : E[Y] = p*n = 0.75*4095 = 3071.25
d) Variance of Y: V[Y] = p(1-p)*n = 0.75 *(1-0.75)* 4095 =767.8125
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