Answer:
a)
![P(X=k) = {15 \choose k} * 0.75^{k}*0.25^{15-k}](https://tex.z-dn.net/?f=%20P%28X%3Dk%29%20%3D%20%7B15%20%5Cchoose%20k%7D%20%2A%200.75%5E%7Bk%7D%2A0.25%5E%7B15-k%7D%20)
For any integer k between 0 and 15, and 0 for other values of k.
b)
![P(X>10) = 0.2252+ 0.2252+ 0.1559+0.0668+0.0134 = 0.6865](https://tex.z-dn.net/?f=P%28X%3E10%29%20%3D%200.2252%2B%200.2252%2B%200.1559%2B0.0668%2B0.0134%20%3D%200.6865)
c) P(6 ≤ X ≤ 10) = 0.2737
d) μ = 15*0.75 = 11.25. σ² = 11.25*0.25 = 2.8125
Step-by-step explanation:
X is a binomial random variable with parameters n = 15, p = 0.75. Therefore
a)
![P(X=k) = {15 \choose k} * 0.75^{k}*0.25^{15-k}](https://tex.z-dn.net/?f=%20P%28X%3Dk%29%20%3D%20%7B15%20%5Cchoose%20k%7D%20%2A%200.75%5E%7Bk%7D%2A0.25%5E%7B15-k%7D%20)
For any integer k between 0 and 15, and 0 for other values of k.
b)
P(X>10) = P(X=11) + P(X=12)+ P(X=13)+P(X=14)+P(x=15)
![P(X=11) = {15 \choose 11} * 0.75^{11} * 0.25^4 = 0.2252](https://tex.z-dn.net/?f=P%28X%3D11%29%20%3D%20%7B15%20%5Cchoose%2011%7D%20%2A%200.75%5E%7B11%7D%20%2A%200.25%5E4%20%3D%200.2252)
![P(X=12) = {15 \choose 12} * 0.75^{12} * 0.25^3 = 0.2252](https://tex.z-dn.net/?f=P%28X%3D12%29%20%3D%20%7B15%20%5Cchoose%2012%7D%20%2A%200.75%5E%7B12%7D%20%2A%200.25%5E3%20%3D%200.2252)
![P(X=13) = {15 \choose 13} * 0.75^{13} * 0.25^2 = 0.1559](https://tex.z-dn.net/?f=P%28X%3D13%29%20%3D%20%7B15%20%5Cchoose%2013%7D%20%2A%200.75%5E%7B13%7D%20%2A%200.25%5E2%20%3D%200.1559)
![P(X=14) = {15 \choose 14} * 0.75^{14} * 0.25 = 0.0668](https://tex.z-dn.net/?f=P%28X%3D14%29%20%3D%20%7B15%20%5Cchoose%2014%7D%20%2A%200.75%5E%7B14%7D%20%2A%200.25%20%3D%200.0668)
![P(X=15) = {15 \choose 15} * 0.75^{15} = 0.0134](https://tex.z-dn.net/?f=P%28X%3D15%29%20%3D%20%7B15%20%5Cchoose%2015%7D%20%2A%200.75%5E%7B15%7D%20%3D%200.0134)
Thus,
![P(X>10) = 0.2252+ 0.2252+ 0.1559+0.0668+0.0134 = 0.6865](https://tex.z-dn.net/?f=P%28X%3E10%29%20%3D%200.2252%2B%200.2252%2B%200.1559%2B0.0668%2B0.0134%20%3D%200.6865)
c) P(6 ≤ X ≤ 10) = P(X = 6) + P(X = 7) + P(X = 8) + P(X=9) + P(X=10)
![P(X=6) = {15 \choose 6} * 0.75^{6} * 0.25^9 = 0.0034](https://tex.z-dn.net/?f=P%28X%3D6%29%20%3D%20%7B15%20%5Cchoose%206%7D%20%2A%200.75%5E%7B6%7D%20%2A%200.25%5E9%20%3D%200.0034)
![P(X=7) = {15 \choose 7} * 0.75^{7} * 0.25^8 = 0.0131](https://tex.z-dn.net/?f=P%28X%3D7%29%20%3D%20%7B15%20%5Cchoose%207%7D%20%2A%200.75%5E%7B7%7D%20%2A%200.25%5E8%20%3D%200.0131)
![P(X=8) = {15 \choose 8} * 0.75^{8} * 0.25^7 = 0.0393](https://tex.z-dn.net/?f=P%28X%3D8%29%20%3D%20%7B15%20%5Cchoose%208%7D%20%2A%200.75%5E%7B8%7D%20%2A%200.25%5E7%20%3D%200.0393)
![P(X=9) = {15 \choose 9} * 0.75^{9} * 0.25^6 = 0.0918](https://tex.z-dn.net/?f=P%28X%3D9%29%20%3D%20%7B15%20%5Cchoose%209%7D%20%2A%200.75%5E%7B9%7D%20%2A%200.25%5E6%20%3D%200.0918)
![P(X=10) = {15 \choose 10} * 0.75^{10} * 0.25^{5} = 0.1652](https://tex.z-dn.net/?f=P%28X%3D10%29%20%3D%20%7B15%20%5Cchoose%2010%7D%20%2A%200.75%5E%7B10%7D%20%2A%200.25%5E%7B5%7D%20%3D%200.1652)
Thereofre,
![P(6 \leq X \leq 10) = 0.0034 + 0.0134 + 0.0393 + 0.0918 + 0.1652 = 0.2737](https://tex.z-dn.net/?f=P%286%20%5Cleq%20X%20%5Cleq%2010%29%20%3D%200.0034%20%2B%200.0134%20%2B%200.0393%20%2B%200.0918%20%2B%200.1652%20%3D%200.2737)
d) μ = n*p = 15*0.75 = 11.25
σ² = np(1-p) = 11.25*0.25 = 2.8125