Question

a company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. over the years, they have established the following probability distribution. let ( x = ) the number of years a new hire will stay with the company. let ( p(x) = ) the probability that a new hire will stay with the company ( x ) years. below is the probability distribution for ( x ). a) fill in the missing probability

( x )	( p(x) )
1	0.18
2	0.29
3	0.16
4
5	0.12
6	0.05

b) what is ( p(x leq 2) )?
c) what is ( p(x>4) )?
d). what is ( p(x = 5) )?
e). on average, how long would you expect a new hire to stay with the company? (calculate the expected value, ( mu ) )

Explanation:

Step1: Find the missing probability for \(x = 4\)

Since the sum of all probabilities in a probability distribution is \(1\). Let \(P(4)=p\). Then \(0.12 + 0.18+0.29 + 0.16+p + 0.12+0.05=1\).

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Step2: Calculate \(P(X\leq2)\)

\(P(X\leq2)=P(0)+P(1)+P(2)\)

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Step3: Calculate \(P(X > 4)\)

\(P(X>4)=P(5)+P(6)\)

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Step4: Calculate \(P(X = 5)\)

From the table, \(P(X = 5)=0.12\)

Step5: Calculate the expected value \(\mu\)

The formula for the expected value of a discrete random variable is \(\mu=\sum_{x}x\cdot P(x)\)

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Answer:

a) \(P(4) = 0.08\)
b) \(0.59\)
c) \(0.17\)
d) \(0.12\)
e) \(\mu=2.46\)