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.
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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