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Exercise 4.12. Suppose our sample space is the interval [0,5), with the uniform prob- ability mea...
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outpu...
Please answer the question thoroughly.
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outputs -2 if the chosen number falls in the interval [0, 1/4). It outputs 1 if the chosen number falls in the interval [1/2,2/3). Otherwise, the random variable iply outputs the chosen number. Find the distribution function Fx associated to X, find its discrete and continuous parts, Fxd and Fxe, and draw their graphs.
Exercise 4.10. A number is...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c),...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c), which of the following is CORRECT? A. E[X] = 0. B. P[X S c/2] = 1. C. Fx(c/2) = 0.5. D. fx(c/2) <fx©). 8. Assuming X is a continuous random variable, which of the following is INCORRECT? A x(x) = dFx(x)/dx. B. x(x) dx = 1. C fx(x) dx = Fzco). D. none of the above. 9. The CLT states that for Z =...
X is a positive continuous random variable with density fX(x). Y = ln(X). Find the cumulative...
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
Question 3 Suppose that the random variable X has the following prob- ability density function. f(x)...
Question 3 Suppose that the random variable X has the following prob- ability density function. f(x) =1- for 1 € (0,2), and zero otherwise. a) Plot the graph of the pdf of x. b) Is it true that Pr[X <0] = 0.5? c) Is it true that Pr[X < 1] = 1? d) Is it true that E[X] < 0.5? e) Is it true that Prix < 0.5) > 0.5? f) Find the CDF of x. Compare the graph of...
5. A continuous random variable X follows a uniform distribution over the interval [0, 8]. (a)...
5. A continuous random variable X follows a uniform distribution over the interval [0, 8]. (a) Find P(X> 3). (b) Instead of following a uniform distribution, suppose that X assumes values in the interval [0, 8) according to the probability density function pictured to the right. What is h the value of h? Find P(x > 3). HINT: The area of a triangle is base x height. 2 0 0
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities...
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities over the events are given by P(cu 1)-0.2, P(ω2)-0.3, P(ag)-0.1, P(04)-0.4 Define the random variable X1 by and the random variable X2 by X2(2) 5, (a) Find the probability distribution of X1 (b) Find the probability distribution of the random variable X1 +X2 Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 0.8, determine K (b)...
1) 2) 3) 4) 5) Suppose that X is a uniform random variable on the interval...
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Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
1. Properties of the uniform, normal, and exponential distributions Aa Aa Suppose that x1 is a...
1. Properties of the uniform, normal, and exponential distributions Aa Aa Suppose that x1 is a uniformly distributed random variable, x2 is a normally distributed random variable, and x3 is an exponentially distributed random variable. For each of the following statements, indicate whether it applies to x1, x2, and/or x3. Check all that apply. x2 x3 (uniform) (normal) (exponential) The area under the graph of the probability density function to the right of the mean equals 0.5. Probabilities are given...
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2)...
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2) and Y = X4 (a) Compute P[Y St] as a function of t. You need to distinguish three different cases. (b) Find the probability density function of Y and use it to compute EY).
4. Suppose X has a discrete uniform distribution: the distribution function of X 5. A random...
4. Suppose X has a discrete uniform distribution: the distribution function of X 5. A random variable Z has the pmf bclow. P (X-х,)-1 , is|2 n. Find 0 Pz(z) 0.20 0.16 0.4 a (1) What is thevalue of a ? (2) What is P(l S Z <3)? (3) What is Fz (1.7)? 6.
Please answer the question thoroughly.
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outputs -2 if the chosen number falls in the interval [0, 1/4). It outputs 1 if the chosen number falls in the interval [1/2,2/3). Otherwise, the random variable iply outputs the chosen number. Find the distribution function Fx associated to X, find its discrete and continuous parts, Fxd and Fxe, and draw their graphs.
Exercise 4.10. A number is...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c), which of the following is CORRECT? A. E[X] = 0. B. P[X S c/2] = 1. C. Fx(c/2) = 0.5. D. fx(c/2) <fx©). 8. Assuming X is a continuous random variable, which of the following is INCORRECT? A x(x) = dFx(x)/dx. B. x(x) dx = 1. C fx(x) dx = Fzco). D. none of the above. 9. The CLT states that for Z =...
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
Question 3 Suppose that the random variable X has the following prob- ability density function. f(x) =1- for 1 € (0,2), and zero otherwise. a) Plot the graph of the pdf of x. b) Is it true that Pr[X <0] = 0.5? c) Is it true that Pr[X < 1] = 1? d) Is it true that E[X] < 0.5? e) Is it true that Prix < 0.5) > 0.5? f) Find the CDF of x. Compare the graph of...
5. A continuous random variable X follows a uniform distribution over the interval [0, 8]. (a) Find P(X> 3). (b) Instead of following a uniform distribution, suppose that X assumes values in the interval [0, 8) according to the probability density function pictured to the right. What is h the value of h? Find P(x > 3). HINT: The area of a triangle is base x height. 2 0 0
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities over the events are given by P(cu 1)-0.2, P(ω2)-0.3, P(ag)-0.1, P(04)-0.4 Define the random variable X1 by and the random variable X2 by X2(2) 5, (a) Find the probability distribution of X1 (b) Find the probability distribution of the random variable X1 +X2 Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 0.8, determine K (b)...
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Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
1. Properties of the uniform, normal, and exponential distributions Aa Aa Suppose that x1 is a uniformly distributed random variable, x2 is a normally distributed random variable, and x3 is an exponentially distributed random variable. For each of the following statements, indicate whether it applies to x1, x2, and/or x3. Check all that apply. x2 x3 (uniform) (normal) (exponential) The area under the graph of the probability density function to the right of the mean equals 0.5. Probabilities are given...
lo (P15) Suppose X is a random variable with the uniform distribution over the interval (1.2) and Y = X4 (a) Compute P[Y St] as a function of t. You need to distinguish three different cases. (b) Find the probability density function of Y and use it to compute EY).
4. Suppose X has a discrete uniform distribution: the distribution function of X 5. A random variable Z has the pmf bclow. P (X-х,)-1 , is|2 n. Find 0 Pz(z) 0.20 0.16 0.4 a (1) What is thevalue of a ? (2) What is P(l S Z <3)? (3) What is Fz (1.7)? 6.
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