Question

Please help me with both these questions below! Thanks in advance.

1. Toss a fair coin for three times and let X be the number of heads.

(a) (4 points) Calculate the expected value of X.

(b) (4 points) Calculate the standard deviation of X?

2. A company that produces fine crystal knows from experience that 1 out of 20 of its goblets have cosmetic flaws and must be classified as “seconds”. Randomly select eight of its goblets, and let X be the number of goblets that are “seconds”.

(a) (2 points) What probability distribution does X follow? Write down the name of the distribution family and state the parameter(s).

(b) (2 points) Calculate P (X = 5).

(c) (2 points) Calculate P (2 ≤ X < 5).

(d) (2 points) Calculate the expected value of X.

(e) (2 points) Calculate the standard deviation of X.

Answer 1

1)

P(head in fair coin) =p = 0.5

here, coin is tossed 3 times, n=3

X can take values 0, 1 , 2 , 3

it is a binomial probability distribution,

and probability is given by

P(X=x) = C(n,x)*px*(1-p)(n-x)

P ( X =    0   ) = C(   3   ,   0   )*   0.50   ^   0   *   0.50   ^   3   =   0.125

P ( X =    1   ) = C(   3   ,   1   )*   0.50   ^   1   *   0.50   ^   2   =   0.375

P ( X =    2   ) = C(   3   ,   2   )*   0.50   ^   2   *   0.50   ^   1   =   0.375

P ( X =    3   ) = C(   3   ,   3   )*   0.50   ^   3   *   0.50   ^   0   =   0.125

X	P(X)	X*P(X)	X² * P(X)
0	0.1250	0	0
1	0.3750	0.375	0.375
2	0.3750	0.75	1.5
3	0.1250	0.375	1.125

	P(X)	X*P(X)	X² * P(X)
total sum =	1	1.5	3

a)

expected value of X = mean = E[X] = Σx*P(X) =            1.5

b)

E [ X² ] = ΣX² * P(X) =            3
          
variance = E[ X² ] - (E[ X ])² =            0.75
          
std dev = √(variance) =            0.8660

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