Question

In: Statistics and Probability

(a) Let X be a continuous random variable which only takes on positive values on the...

(a) Let X be a continuous random variable which only takes on positive values on the interval [1, 4]. If P(X) = (√ x + √ 1 x )C 2 for all x in this interval, compute the value of C.
(b) Let X be a random variable with normal distribution. Let z represent the z-score for X, and let a be a positive number. Prove that P(z < |a|) = P(z < a) + P(z > −a) − 1.

Solutions

Expert Solution

Explanation:-

Probability density function

Let f(x) be any continuous function of x satisfies the following conditions

i)   

ii)

iii)

Let 'X' be a continuous random variable which only takes on positive values on the interval [1,4]

Given function

Since the total probability is unity

By using integration formula

Now  

now integrating , we get

  

on simplification , we get

On calculation , we get

b)  

Given 'X' be a random variable with Normal distribution and Let Z be the score of X

Hence


Related Solutions

Let X be a continuous random variable such that E[Xm] exists where m is some positive...
Let X be a continuous random variable such that E[Xm] exists where m is some positive integer. Prove that if k is a positive integer and k < m, then E[Xk] exists.
1. Let X be a continuous random variable such that when x = 10, z =...
1. Let X be a continuous random variable such that when x = 10, z = 0.5. This z-score tells us that x = 10 is less than the mean of X. Select one: True False 2. If an economist wants to determine if there is evidence that the average household income in a community is different from $ 32,000, then a two-tailed hypothesis test should be used. Select one: True False 3. α (alpha) refers to the proportion of...
Let x be a continuous random variable that follows a distribution skewed to the left with...
Let x be a continuous random variable that follows a distribution skewed to the left with ?= 92 and ?=15. Assuming n/N <= .05, find the probability that the sample mean, x bar, for a random sample of 62 taken from this population will be (ROUND ANSWERS TO FOUR DECIMAL PLACES): a) less than 81.5 P(less that 81.5)= b) greater than 89.7 P(greater than 89.7)= Please show your work.
Let x be a continuous random variable that is normally distributed with a mean of 65...
Let x be a continuous random variable that is normally distributed with a mean of 65 and a standard deviation of 15. Find the probability that x assumes a value: less than 48 greater than 87             between 56 and 70
Let X be a continuous random variable with a probability density function fX (x) = 2xI...
Let X be a continuous random variable with a probability density function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) = e −x a. Find the expression for the probability density function fY (y). b. Find the domain of the probability density function fY (y).
Let W be a random variable that takes values 1 to 6 with equal probability. a)...
Let W be a random variable that takes values 1 to 6 with equal probability. a) Write the PMF. 
 b) Calculate the mean and variance of W and z=10 W.
Let 'x' be a random variable that represents the length of time it takes a student...
Let 'x' be a random variable that represents the length of time it takes a student to write a term paper for Dr. Adam's Sociology class. After interviewing many students, it was found that 'x' has an approximately normal distribution with a mean of µ = 7.3 hours and standard deviation of ơ = 0.8 hours. For parts a, b, c, Convert each of the following x intervals to standardized z intervals. a.) x < 8.1   z < b.) x...
2] Let x be a continuous random variable that has a normal distribution with μ =...
2] Let x be a continuous random variable that has a normal distribution with μ = 48 and σ = 8 . Assuming n N ≤ 0.05 , find the probability that the sample mean, x ¯ , for a random sample of 16 taken from this population will be between 49.64 and 52.60 . Round your answer to four decimal places.
Let x be a continuous random variable that has a normal distribution with μ = 60...
Let x be a continuous random variable that has a normal distribution with μ = 60 and σ = 12. Assuming n ≤ 0.05N, where n = sample size and N = population size, find the probability that the sample mean, x¯, for a random sample of 24 taken from this population will be between 54.91 and 61.79. Let x be a continuous random variable that has a normal distribution with μ = 60 and σ = 12. Assuming n...
Let X be a continuous random variable following normal distribution with mean value of: (a is...
Let X be a continuous random variable following normal distribution with mean value of: (a is 1) and standard deviation of b is 1/10 ,  What is the mode of X? (1 mark)  What is median of X? (1 mark)  What is ?(? > ?)? (1mark)  What is ?(? − ? < ? < ? + ?)? (1 mark)  What is ?(? − 1.96? < ? < ? + 1.96?))? (1mark)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT