The probability distribution of a discrete random variable x is shown below, whe
ID: 3204863 • Letter: T
Question
The probability distribution of a discrete random variable x is shown below, where x represents the number of motorcycles owned by families in the state of Michigan. Find the E(X) and VOX) of this distribution. Find the mean and the variance of Y where Y = 5X + 3. Suppose that 40percentage of all students who have to buy a text for a particular course want a new copy (the successes!), whereas other 60percentage want a used copy. Consider randomly selecting. Note that the number of students among these 7 purchasers who buy a new textbook follows a binomial distribution (X - Bio(7.04). What are the expected value and variance of this distribution? what probably that at least one student among these 7 purchasers will buy a new textbook? Births in a hospital occur randomly at an average rate of 5 births per hour. Assuming that the number of births follow a Poisson distribution: Find the variance of the number of births in one hour. Find the probability that there will be no more than 2 birth in a 30-minute period.Explanation / Answer
2. a. Mean, mu=E(X)=sigma x*P(x), where, x denotes the values for random variable X, and P(x) denotes corresponding probabilities.
=0*0.1+1*0.50+2*0.10+3*0.3
=1.6 (ans)
Variance, Var(X)=sigma (x-mu)^2*P(x)
=(0-1.6)^2*0.1+(1-1.6)^2*0.50+(2-1.6)^2*0.10+(3-1.6)^2*0.3
=0.784 (ans)
b. Given Y=5X+3
Therefore, E(Y)=E(5X+3)
=5E(X)+3
Substitute the value of E(X).
=5*1.6+3 [Applying rule, E(X+-c)=E(X)+-c]
=11 (ans)
Var(Y)=Var(5X)+3 [ Applying rule, Var(X+-c)=Var(X)]
=5^2Var(X) [Applying rule, Var(aX)=a^2 Var(X)]
=25*0.784
=19.6 (ans)
2. a. Given the distribution is binomial in nature, with p=0.4, and n=7.
Therefore, mean, mu=E(X)=np=7*0.4=2.8 (ans)
Var(X)=npq, where, q denotes probability of failure.
=7*0.4*0.6
=1.68 (ans)
b. P(X>=1)=1-P(X<1)=1-P(X=0)
=1-7C0(0.4)^0(0.6)^7 [Using formula for binomial distribution, P(X,r)=nCr(p)^r(1-p)^n-r, where, n is number of trials, r is specific number of success in n trials, and p denote probability of success]
=1-0.0280
=0.9720 (ans)
3. a. The mean birth rat eis 5 births/hour, converting into 30-minute, the birth rate becomes 2.5. Poisson distribution has, mean=variance, therefore, variance is: 2.5. (ans)
b. P(X<=2)=P(X=0)+P(X=1)+P(X=2)
=2.718^(-2.5)(2.5)^0/0!+2.718^(-2.5)(2.5)^1/1!+2.718^(-2.5)(2.5)^2/2! [Applying formula, P(X,mu)=e^(-mu)(mu)^x/x!, where, x denotes specific number of success within a region, mu is mean number of success in a specified region, and e=2.718]
=0.5438 (ans)
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