Find the characteristic function of a random variable having the binomial distribution with parameters p and n.
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- A fair standard die is rolled 50 times. Let W be a random variable with binomial distribution thát represents the number of times the face with a six on it appears uppermost. Pr(W = k+1) _ (50 – k) 5(k +1) b. Show that %3D Pr(W = k)the height of a randomly selected male individual in a population is a normal random variable with mathematical expectation m=178 σ =8 what % of the individuals of this population is taller than 185cm?the height of the female individuals in this population is of normal distribution with parameters m=165 cm and σ=7cm. what % of the women are taller than half of the menElizabeth is a busy pediatrician. On any given day, she diagnoses an average of five babies with middle-ear infections. Assume that the number of babies who come to her clinic with middle-ear infections is a Poisson random variable. Calculate the probability that fewer than three babies with middle-ear infections will come to her clinic tomorrow. Give your answer in decimal form precise to three decimal places. P(X < 3) =
- Suppose 500 people audition for admission to a dance school. Each dancer in one half of the group will be admitted with probability 0.07, while each dancer in the other half of the group will be admitted with probability 0.04. Let X be the total number of dancers admitted. (Notice that X is the sum of two binomial variables). (a) Find the mean and variance of X. (b) Use normal approximations to binomial random variables to approximate (i) the probability that more than 30 dancers are admitted. (ii) the probability that at least 20 dancers are admitted.The service time for a passport application in a certain government office is modelled as an exponential random variable with parameter λ = 5 per hour, independent and identically distributed for all applicants. Determine the probability that 50 applicants is accommodated within eight hours. Determine the maximum number of applicants such that the chance of accommodating all of them within eight hours is at least 95%.The expected life of a certain component in a mechanism is treated as a random variable having a gamma distribution with a = 3 and ß = 2. The average expected life of such component is 12 months, what is the probability that this component can have a greater than average expected life?
- Weights of Lamb after eight weeks after born are normally distributed with mean µ = 14 kg and variance σ 2 = 4. a)Find the probability p that a single lamb has weight less than 12 kg. b) Find the probability that exactly 3 of the next 5 lambs examined will have weights less than 12 kg.The number of automobile sold weekly at a certain dealership is a random variable with expected value 16.Give an upper bound to the probability that the next week sales exceed 18.Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find P ( X = 5 )
- Provide an example of a binomial random variable and explain how each condition for the binomial distribution is fulfilled.A certain type of tree has seedlings randomly dispersed in a large area, with the mean density of seedlings being approximately three per square yard. If the seedlings are randomly dispersed, the number of seedlings per region, Y can be modelled as a Poisson random variable. If a 1 forester randomly locates ten 1-square-yard sampling regions in the area, the probability that none of the regions will contain seedlings is 0.0498. 2.3.1 If the seedlings really are randomly dispersed, the number of seedlings per region, Y, can be modelled as a Poisson random variable with 2 = 3. Interpret 1 =3. 2.3.2 State the moment generating function_of the random variable Y.Use the normal approximation to the binomial to calculate the probability of getting more than 39 defectives in a random sample of 400 taken from a population which is 8% defective.
