Answer:
a) 0.9
b) Mean = 1.58
Standard Deviation = 0.89
Step-by-step explanation:
We are given the following in the question:
A marketing firm is considering making up to three new hires.
Let X be the variable describing the number of hiring in the company.
Thus, x can take values 0,1 ,2 and 3.
![P(x\geq 2) = 50\%= 0.5\\P(x = 0) = 10\% = 0.1\\P(x = 3) = 18\% = 0.18](https://tex.z-dn.net/?f=P%28x%5Cgeq%202%29%20%3D%2050%5C%25%3D%200.5%5C%5CP%28x%20%3D%200%29%20%3D%2010%5C%25%20%3D%200.1%5C%5CP%28x%20%3D%203%29%20%3D%2018%5C%25%20%3D%200.18)
a) P(firm will make at least one hire)
![P(x\geq 2) = P(x=2) + P(x=3)\\0.5 = P(x=2) + 0.18\\ P(x=2) = 0.32](https://tex.z-dn.net/?f=P%28x%5Cgeq%202%29%20%3D%20P%28x%3D2%29%20%2B%20P%28x%3D3%29%5C%5C0.5%20%3D%20P%28x%3D2%29%20%2B%200.18%5C%5C%20P%28x%3D2%29%20%3D%200.32)
Also,
![P(x= 0) +P(x= 1) + P(x= 2) + P(x= 3) = 1\\ 0.1 + P(x= 1) + 0.32 + 0.18 = 1\\ P(x= 1) = 1- (0.1+0.32+0.18) = 0.4](https://tex.z-dn.net/?f=P%28x%3D%200%29%20%2BP%28x%3D%201%29%20%2B%20P%28x%3D%202%29%20%2B%20P%28x%3D%203%29%20%3D%201%5C%5C%200.1%20%2B%20P%28x%3D%201%29%20%2B%200.32%20%2B%200.18%20%3D%201%5C%5C%20P%28x%3D%201%29%20%3D%201-%20%280.1%2B0.32%2B0.18%29%20%3D%200.4)
![\text{P(firm will make at least one hire)}\\= P(x\geq 1)\\=P(x=1) + P(x=2) + P(x=3)\\ = 0.4 + 0.32 + 0.18 = 0.9](https://tex.z-dn.net/?f=%5Ctext%7BP%28firm%20will%20make%20at%20least%20one%20hire%29%7D%5C%5C%3D%20P%28x%5Cgeq%201%29%5C%5C%3DP%28x%3D1%29%20%2B%20P%28x%3D2%29%20%2B%20P%28x%3D3%29%5C%5C%20%3D%200.4%20%2B%200.32%20%2B%200.18%20%3D%200.9)
b) expected value and the standard deviation of the number of hires.
![E(x^2) = \displaystyle\sum x_i^2P(x_i)\\=0(0.1) + 1(0.4) + 4(0.32) +9(0.18) = 3.3\\V(x) = E(x^2)-[E(x)]^2 = 3.3-(1.58)^2 = 0.80\\\text{Standard Deviation} = \sqrt{V(x)} = \sqrt{0.8036} = 0.89](https://tex.z-dn.net/?f=E%28x%5E2%29%20%3D%20%5Cdisplaystyle%5Csum%20x_i%5E2P%28x_i%29%5C%5C%3D0%280.1%29%20%2B%201%280.4%29%20%2B%204%280.32%29%20%2B9%280.18%29%20%3D%203.3%5C%5CV%28x%29%20%3D%20E%28x%5E2%29-%5BE%28x%29%5D%5E2%20%3D%203.3-%281.58%29%5E2%20%3D%200.80%5C%5C%5Ctext%7BStandard%20Deviation%7D%20%3D%20%5Csqrt%7BV%28x%29%7D%20%3D%20%5Csqrt%7B0.8036%7D%20%3D%200.89)