Respuesta :
Answer:
(a) The mean, variance and standard deviation of X are 1.60, 0.96 and 0.98 respectively.
(b) The mean, variance and standard deviation of X are 3.20, 0.64 and 0.80 respectively.
(c) The mean, variance and standard deviation of X are 1.50, 0.75 and 0.87 respectively.
(d) The mean, variance and standard deviation of X are 4.00, 0.80 and 0.89 respectively.
Step-by-step explanation:
The random variable X follows a Binomial distribution with parameters n and p.
The mean, variance and standard deviation of X are:
[tex]\mu=np\\\sigma^{2}=np(1-p)\\\sigma=\sqrt{np(1-p)}[/tex]
(a)
For n = 4 and p = 0.40 compute the mean, variance and standard deviation of X as follows:
[tex]\mu=np=4\times0.40=1.60\\\sigma^{2}=np(1-p)=4\times0.40\times(1-0.40)=0.96\\\sigma=\sqrt{np(1-p)}=\sqrt{0.96}=0.98[/tex]
Thus, the mean, variance and standard deviation of X are 1.60, 0.96 and 0.98 respectively.
(b)
For n = 4 and p = 0.80 compute the mean, variance and standard deviation of X as follows:
[tex]\mu=np=4\times0.80=3.20\\\sigma^{2}=np(1-p)=4\times0.80\times(1-0.80)=0.64\\\sigma=\sqrt{np(1-p)}=\sqrt{0.64}=0.80[/tex]
Thus, the mean, variance and standard deviation of X are 3.20, 0.64 and 0.80 respectively.
(c)
For n = 3 and p = 0.50 compute the mean, variance and standard deviation of X as follows:
[tex]\mu=np=3\times0.50=1.50\\\sigma^{2}=np(1-p)=3\times0.50\times(1-0.50)=0.75\\\sigma=\sqrt{np(1-p)}=\sqrt{0.75}=0.87[/tex]
Thus, the mean, variance and standard deviation of X are 1.50, 0.75 and 0.87 respectively.
(d)
For n = 5 and p = 0.80 compute the mean, variance and standard deviation of X as follows:
[tex]\mu=np=5\times0.80=4.00\\\sigma^{2}=np(1-p)=5\times0.80\times(1-0.80)=0.80\\\sigma=\sqrt{np(1-p)}=\sqrt{0.80}=0.89[/tex]
Thus, the mean, variance and standard deviation of X are 4.00, 0.80 and 0.89 respectively.
