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Suppose the number of automobile accidents a driver will be involved in during a one-year period is a random variable X having a Poisson distribution with parameter θ ,where θ is a measure of accidents proneness that varies from driver to driver in accordance with Gamma distribution given by
f(θ) = (p/q)^α × θ^(α−1)/Γ(α)×exp {(−p/q) θ} , θ > 0
where α is a positive integer, p, q are positive constants and p + q = 1.
(a) show that he factorial moment generating function of x is g(s) = (p/1 − qs)^α
.
(b) Using the uniqueness property of probability generating functions, identify completely the
distribution of X.
(c) If p =2/3 and α = 12, find
E{x(x − 1)· · ·(α − k + 1)},
where k is a positive integer
Random sample of n=2 are drawn from a finite population consisting of the numbers 5,6,7,8,and 9
A.Find the mean population
B.find the standard deviation of the population
C. Find the mean of the sampling distribution of the sample means.
D.FIND The standard deviation of the sampling distribution of the sample means
E.Verify the central limit theorem
A coordinator of a degree program in a university assessed a probability distribution for the number of students (x) entering into the program as follows:
X 30 45 60 65 90 100
f(x) 0.10 0.20 0.20 0.30 0.10 0.10
(i) Is the probability distribution valid? Explain
(ii) What is the probability of 60 or fewer students entering into the program?
A3. Suppose that the life span in months of a lead bulb is a random variable and let Y1 and Y2 denote the life span of two different types of lead bulbs produced by different companies. If Y1 and Y2 are independent exponentially distributed random variables, both with mean β and let X1 = Y1/Y1 + Y2 and X2 = Y1 + Y2. Find and identify the marginal distributions of X1 and X2
A research firm conducted a study in 2014 and found that 40% of Internet users
received more than 10 emails per day. A similar study on the use of email
repeated in 2016.
i. Formulate the hypotheses that can be used to determine whether the
proportion of Internet users receiving more than 10 emails per day
increased in 2016.
i i. In a sample of 450 Internet users found 190 receiving more than 10 emails per
day, what is P-value?
iii. At a = 0.05, what is your conclusion?
(C) Consider the following hypothesis test:
Ho; = 10
H A : not equal to10
A sample of 50 provided a sample mean of 9.80. The population standard deviation is
4.
i. Compute the value of the test statistic.
i i. What is P-value?
iii. At a = 0.10, what is your conclusion?
Suppose that the continuous random variable x has the probability density function
f(x) = 1/2e−|x| , −∞ < x < ∞
find the m.g.f of x and use it to find E(x) and V ar(x).
Find the following probabilities if the random variable X is normally distributed with
mean 15 and standard deviation 3.
(i) P(X<10)
(ii) P(10<X<25 )
Given that Z is a standard normal random variable, find z for each situation,
(i) The area to the left of Z is 0.2115
(ii) The area to the right of Z is 0.5000
The average amount parents spent per child on private tuition classes is Rs 6000.
Assume that the standard deviation is Rs 1000 and that the amount spent is normally
distributed.
(i) Find the probability that the amount spent on a randomly selected child is:
a. Less than Rs 5000
b. Between Rs 5000 and Rs 6500
c. What is the probability that the amount spent on a randomly selected
child is between Rs 6500 and RS 8000. (06 marks)
(ii) Assume that 5% of parents spent over Rs 8000. If we selected randomly 1000
parents who sent their children to tuition, how many parents would spend more
than Rs 8000?
1/ A production manager wishes to examine if there is a significant difference in the number of production units produced by the night shift and the day shift. A random sample of night and day shifts was selected, and the number of production units for each shift was recorded. The following results were obtained:
Mean S N
Day shift 23.4 7.2 70
Night shift 19.5 6.3 70
Using a 5% significance level, present your conclusion.
the score of shs students in their first quarter in statistic and probability brainly were analyzed and found to have a mean of 54.5and standard deviation of 7.5. find a 90% confidence interval
