Let X be a continuous random variable with probability density function fx (x) = = {1/³ if -1 < x < 2 otherwise Let Y = X². Find the cumulative distribution function for Y.
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- Let X be a random variable with density function 1) ={0. k(1 -x), if 0<*<1; otherwise. f(x) Find k, together with the expectation and the variance of the random variable Y defined as Y = 3X - 1.5. The probability distribution of discrete random variable X is given by k for x = 1, 2, 3 x +1 P(X = x) %3DUsing the Gaussian Probability Density function write a function that will calculate the probability that variable x will be in the range µ − ασ < x < µ + ασ, where α is some multiple of the variance.
- X is a discrete random variable with the following PMF: P(X = 1.4) = 0.25 P(X = -9.7) = 0.25 P(X = 10) = 0.5 Find the standard deviation of X.The sample space of a random experiment is (a, b, c, d, e, f, and each outcome is equally likely. A random variable is defined as follows: outcome X с d TEISEER 1.5 1.5 a b 0 2 Determine the probability mass function of X. Use the proba- bility mass function to determine the following probabilities: (a) P(X= 1.5) (b) P(0.5 3) (d) P(0 ≤X<2) (e) P(X=0 or X = 2)Suppose the following is a valid probability distribution. Find the value of m. 3 4 6. P(x) 0.05 0.12 0.35 0.32 Select one: a. 0.16 b. cannot be determined C. 0.08 d. 0.84 pe here to search 近
- 7. Let X be a random variable with expectation EX <∞o. Prove that Var(X) = E(X²) – E(X)².For a given joint probability distribution f (x, y) where 0 < x < 4 and 0 < y < 5, which of the following equation provides the marginal probability of y? O f f (x, y) dy O S f (x, y) dx S f (x, y) dx O sf f (x, y) dæLet X and Y be two binary, discrete random variables with the following joint probability mass functions. (a) Compute P(X = 0]Y = 1). (b) Show that X and Y are not statistically independent. P(X = 0, y = 1) = P(X = 1, Y = 0) = 3/8 P(X = 0, y = 0) = P(X = 1, Y = 1) = 1/8
- Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ=1/10. If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait, (a) More than 10 minutes (b) Between 10 and 20 minutesSolve In R programmning language: Calculate the probability for each of the following events: (a) A standard normally distributed variable is less than -2.5. (b) A normally distributed variable with mean 35 and standard deviation 6 is larger than 42 but less than 45. (c) A normally distributed variable with mean 35 and standard deviation 6 is larger than 40 but less than 41. (d) X < 0.9 when X has the standard uniform distribution (min=0, max=1). (e) 1 < X < 3 in the exp distribution with rate λ = 2.2. Let D be a distribution over R where the mean is 5 and variance is 9. Suppose x1, ... , x10 are indepen- dent draws from D. Plot the possible positions of these random variables on the real line. 3. Let w E Rd be the variable, and let x E Rd and y E R be given. Calculate the gradient of the following functions with respect to w: • F(w) = (y – w · x)100; • F(w) = vrwsi 1 y+w.x' • F(w) = log(1+ yw · x); F(w) = e(w-x)².