Question

QUESTION 25

Suppose that X is a binomial random variable with n D 25 and p D 0:5: Calculate

(a) the probability P .X D 15/: (2)

(b) the probability P .X  16/: (4)

(c) the expected value of X. (2)

(d) the variance of X. (3)

Answer 1

$X\sim Bin (25, 0.5)$

(a)

$$P(X=15)=\dbinom{25}{15}(0.5)^{15}(1-0.5)^{25-15}$$

$$=0.09741663933$$

(b)

$$P(X>16)=P(X=17)+P(X=18)$$

$$+P(X=19)+P(X=20)$$

$$+P(X=21)+P(X=22)$$

$$+P(X=23)+P(X=24)$$

$$+P(X=25)$$

$$=\dbinom{25}{17}(0.5)^{17}(1-0.5)^{25-17}$$

$$+\dbinom{25}{18}(0.5)^{18}(1-0.5)^{25-18}$$

$$+\dbinom{25}{19}(0.5)^{19}(1-0.5)^{25-19}$$

$$+\dbinom{25}{20}(0.5)^{25}(1-0.5)^{25-20}$$

$$+\dbinom{25}{21}(0.5)^{21}(1-0.5)^{25-21}$$

$$+\dbinom{25}{22}(0.5)^{22}(1-0.5)^{25-22}$$

$$+\dbinom{25}{23}(0.5)^{23}(1-0.5)^{25-23}$$

$$+\dbinom{25}{24}(0.5)^{23}(1-0.5)^{25-24}$$

$$+\dbinom{25}{25}(0.5)^{23}(1-0.5)^{25-25}$$

$$=0.05387607217$$

$$P(X\ge16)=P(X=16)+P(X>16)$$

$$=\dbinom{25}{16}(0.5)^{16}(1-0.5)^{25-16}+P(X>16)$$

$$=0.11476147175$$

(c)

$$E(X)=np=25(0.5)=12.5$$

(d)

$$Var(X)=np(1-p)=25(0.5)(1-0.5)=6.25$$