A box contains four new, one used but still working, and two defective diodes. Five diodes are randomly taken out from that box. Find the joint probability mass function p(4, 1), which describes the probability to take out from the box all of the useful diodes at once.
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- The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Let M and W represent the number of men and women, respectively, that enter the store in a given hour. (Assume that each new customer is equally likely to be a man or a woman.) (a) Find the joint probability mass function of M and W. That is, find a formula for p(i, j) = P(M = i, W = j). (Hint: First write p(i, j) = P(M = i, W = j) = P(M = i, M + W = i + j) and then compute the term on the right-hand side by using the conditional probability formula P(A ∩ B) = P(A|B)P(B).) (b) Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour.A survey of cars on a certain stretch of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a randomly chosen car. a) Find the probability mass function of X. b) Find P(X ≤ 2). c) Find P(X > 3). d) Find μX. e) Find σX.The daily demand for pumpkin pies at a corner grocery store is a random variable with probability mass function 1 4 (1) for k= 0, 1,2,3,4 Px (k) = 16 k 0 otherwise The cost of baking one pumpkin pie is $1.00. The store sells the pies for $3.00, each. At the beginning of every day, the store manager must decide how many pies to bake. At the end of the day, any unsold pies are thrown away. If the demand for pies exceeds the supply, you can only sell those pies that you have already baked for that day. The additional customers leave unsatisfied. (a) Find P(X> 0). (b) Find E(X). (c) Find Var(X). (d) If the manager decides to bake three pies, what is the ex- pected profit for that day?
- A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales.A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint probability mass func- tion of N1 and N2.A computer can provide you an infinite sequence of random integer numbers between 1 and 5 with the probability mass function P(X=x)=1/5 for x ∈ {1,2,3,4,5}.Using only this random sequence provided by the computer, generate a random sequence of integer numbers between 1 and 7 with the probability mass functionP(Y=y)=1/7 for y ∈ {1,2,3,4,5,6,7}.
- The time that it takes to service a car is an exponential random variable with rate 1.1. If A brings his car in at time 0 and B brings his car in at time t, what is the probability thatB’s car is ready before A’s? Assume that the service times are independent, and the servicebegins upon arrival.2. If both cars are brought in at time 0, with work starting on B’s car only when A’s car has beencompletely serviced, what is the probability that B’s car is ready before 2? Detailed explanation and calculation would be much appreciated.If λ = 6 per hour and μ = 6 per hour in an (M/M/1): (5/FIFO) model, find the probability that there is no customer in the system.another random variable Y. The joint probability masS function of X and Y is dter random variable Y. The joint probability mass function of X and Y is listed below. am put to a communication channelis a random variable X and the output is Y/X -1 1/4 1/8 -1 Р (Х, Y) %3D 1/4 1 1/8 1/4 Find (a) P (Y=1/X=1) (b) P (X=1/Y=1)