Qn.2. (a) A continuous random variable x has the pdf, f(x) = 2 xk 0< x < 1 and 0 otherwise find the (i) value of the constant k (ii) mean and standard deviation of X (b) A random variable X takes on the integer value x, with probability, Q (X= x) such that, Q(X= x) = k x3, for x = 1,2,3,4 and 0 otherwise Find (i) value of k (ii)Mean and variance of the distribution (iii) Mean and variance of 2X- 4
Question
Qn.2. (a) A continuous random variable x has the pdf, f(x) = 2 xk 0< x < 1 and 0 otherwise find the (i) value of the constant k (ii) mean and standard deviation of X (b) A random variable X takes on the integer value x, with probability, Q (X= x) such that, Q(X= x) = k x3, for x = 1,2,3,4 and 0 otherwise Find (i) value of k (ii)Mean and variance of the distribution (iii) Mean and variance of 2X- 4
Solution
Sure, let's solve this step by step.
(a) (i) To find the value of the constant k, we need to know that the total probability should be 1. So, we integrate the pdf from 0 to 1 and set it equal to 1.
∫ from 0 to 1 (2xk) dx = 1 => x^2k from 0 to 1 = 1 => k = 1
(ii) The mean (μ) of a continuous random variable X with pdf f(x) is given by: μ = ∫ from -∞ to ∞ (x * f(x)) dx => μ = ∫ from 0 to 1 (x * 2x) dx => μ = ∫ from 0 to 1 (2x^2) dx => μ = 2/3
The variance (σ^2) is given by: σ^2 = ∫ from -∞ to ∞ ((x - μ)^2 * f(x)) dx => σ^2 = ∫ from 0 to 1 ((x - 2/3)^2 * 2x) dx After calculating this integral, we can find the standard deviation (σ) by taking the square root of the variance.
(b) (i) Similar to part (a), we find the value of k by setting the sum of all probabilities equal to 1. k * (1^3 + 2^3 + 3^3 + 4^3) = 1 => k = 1/100
(ii) The mean (μ) of a discrete random variable X with probability mass function Q(X=x) is given by: μ = ∑ (x * Q(X=x)) => μ = ∑ from x=1 to 4 (x * kx^3) => μ = k * (11^3 + 22^3 + 33^3 + 44^3)
The variance (σ^2) is given by: σ^2 = ∑ ((x - μ)^2 * Q(X=x)) => σ^2 = ∑ from x=1 to 4 ((x - μ)^2 * kx^3) After calculating this sum, we can find the standard deviation (σ) by taking the square root of the variance.
(iii) If Y = 2X - 4, then the mean of Y is E(Y) = 2E(X) - 4 and the variance of Y is Var(Y) = 4Var(X).
Similar Questions
Let X be a continues random variable with PDF (𝑥) = 𝐴𝑒−𝑥3 ; 𝑥 ≥ 0 . Find (i) A (ii) Mean (iii) Variance
The discrete random variable X has the probability functionP (X = x) =kx if x = 2, 4, 6k(x − 2) if x = 80 otherwisewhere k is a constant.(a) Show that k = 118 .(b) Find the exact value of F (5).(c) Find the exact value of E(X).(d) Find the exact value of E(X2).(e) Calculate Var(3 − 4X).1
1. The sum of all probabilities of a discrete random variable X is equal to __________.A. 0 B. 1 C. 2 D. 3______2. A random variable is ____ if its set of possible outcomes is countable.A. discrete B. continuous C. interval D. ratio______3. Which of the following is a continuous random variable?A. Philippine population B. SHS students C. temperature D. daily budget______4. Which of the following is more variable, that is the values of random variable are farther from the mean?A. Var(X) = 1.2 B. Var(X) = 2.01 C. Var(X) = 2.98 D. Var(X) = 3.10______5. Expected value of random variable X is also known as which of the following?A. Standard Deviation B. Variance C. Mean D. Median______6. In a probability distribution, one of the values of random variable X has a negative probability, and thesum of all probabilities is one. Which of the following is True?A. It is a pmf B. It is not a pmf C. Both A and B D. Neither A nor B______7. Which of the following is a discrete random variable?A. weight of grade 1 pupils C. monthly incomeB. volume of water D. acceleration of a car______8. If x is not an element of the Support S of the random variable X, then the probability of that x is ____.A. Negative B. 0 C. 0 < 𝑃(𝑥) < 1 D. 1______9. A player draws a card at random from a standard deck of 52 cards. If the card drawn is red, the playerswins 1,000.00 but if it is black, he loses the same amount. What is the expected value of the game?A. 0 B. 1 C. 2 D. – 1______10. It is a table that gives a list of probability values along with their associated value in the range of adiscrete random variable.A. Probability distribution B. PMF C. Both A and B D. Neither A nor B______11. Which of the following is the area under the normal curve?A. 0 B. 1 C. 2 D. 3______12. The curve is symmetrical to which of the following?A. mean B. variance C. area D. SD______13. If the z-value is 0.42, what is the area?A. 0.0800 B. 0.0279 C. 0.1628 D. 0.3129______14. Find the area if the value of 𝑧 = −0.13.A. 0.0248 B. 0.0517 C. 0.1398 D. 0.2467______15. If the area is 0.4977, what is the z-value?A. 1.63 B. 1.97 C. 2.01 D. 2.83______16. Find the area between 𝑧 = −1 and 𝑧 = −3.A. 0.8300 B. 0.8350 C. 0.8400 D. 0.8450______17. Find the area between 𝑧 = −1.5 and 𝑧 = −1.3.A. 0.0100 B. 0.0200 C. 0.0300 D. 0.0400______18. If the area is 0.8351, what are the two values of z.A. 𝑧 = 0.24 and 𝑧 = 0.13 C. 𝑧 = −0.13 and 𝑧 = −1.3B. 𝑧 = 1.6 and 𝑧 = 2.1 D. 𝑧 = −2.5 and 𝑧 = 1______19. Find the area of the proportion in the probability notation 𝑃(1 < 𝑧 < 3).A. 0.1573 B. 0.1524 C. 0.1575 D. 0.1576______20. Solve for the proportion of the area of the probability notation 𝑃(𝑧 > 1)A. 0.1585 B. 0.1586 C. 0.1587 D. 0.1588______21. Find the area of the proportion in the probability notation 𝑃(> −2.5).A. 0.9937 B. 0.9934 C. 0.9936 D. 0.9938
Consider the random variable X ∼ N (5, 9); that is, a normal random variable with mean µ = 5and standard deviation σ = 3.(a) Evaluate the following:P(X ≤ 2), P(X > 2), P(X ≤ 8), P(X > 8).(b) Find the points x such that:P(X ≤ x) = 0.10, P(X ≤ x) = 0.25, P(X ≤ x) = 0.52, P(X ≤ x) = 0.99.Page 2 Page 2 of 3
A random variable X that assumes the values x1, x2,...,xk is called a discrete uniform random variable if its probability mass function is f(x) = 1 k for all of x1, x2,...,xk and 0 otherwise. Find the mean and variance of X
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.