Let X be a binomial random variable wih n 15 and p 9 a fin

Let X be a binomial random variable wih n = 15 and p = .9

(a) find P(X=13), we found it to be .26691

(b) find P(8 less than or equal to x less than or equal to 14), we found it to be

(c) Compute E(X) and Var (X), I\'m not sure how to compute these from the given information.

Solution

A)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    15      
p = the probability of a success =    0.9      
x = the number of successes =    13      
          
Thus, the probability is          
          
P (    13   ) =    0.266895912 [answer]

b)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    8      
x2 =    14      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.9      
          
Then          
          
P(at most    7   ) =    3.36249E-05
P(at most    14   ) =    0.794108868
          
Thus,          
          
P(between x1 and x2) =    0.794075243   [ANSWER]

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c)

E(x) = n p = 15*0.9 =    13.5 [ANSWER]

Var(x) = np(1-p) = 15*0.9*(1-0.9) = 1.35 [ANSWER]