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- Let X represent a continuous random variable with a Uniform distribution over the interval from 0 to 2. Find the following probabilities (use 2 decimal places for all answers):(a) P(X ≤ 0.22) = (b) P(X < 0.22) = (c) P(0.98 ≤ X ≤ 1.92) = (d) P(X < 0.98 or X > 1.92) =Consider the binomial distribution P(k) = b(k; n, p). Show that: (i) (n-k+ 1)p kq 0. (ii) P(k) > P(k-1) for k (n+1)p. P(k) b(k; n, p) P(k-1) b(k-1; n, p) 1Q1) 14% of the adults in a certain population are infected by a Corona Virus. Five adults are selected randomly from this population for diagnoses. Find the probability that at least two of them will be infected by .this Virus 1 .Q2) Suppose that f(X)=X,12). a) Find .b) Find the mean Q3) A study shows that the systolic blood pressures for adults in a certain population are approximately normally distributed with mean of 115 and .standard deviation of 10 a) Find the probability that a randomly selected person will have a blood -pressure between 109 and 124 Sel).b) Find the reading that is exceeded by only 5% of the population التي يتجاوزها 5% من المجتمع(
- Could you solve (i), (ii), (iii)? Thank you.A company has 6000 arrivals of Internet traffic over a period of 12,160 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these μ*• e -μ to find the probability of X! Internet arrivals have a Poisson distribution. If we want to use the formula P(x) = exactly 3 arrivals in one thousandth of a minute, what are the values of µ, x, and e that would be used in that formula? μ= (Round to three decimal places as needed.) X = e = (Round to three decimal places as needed.) tv -O). ☐☐ + S il 1 √₁ Vi A O (LI) More W Next XAssume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).
- The amount of time it takes Susan to wait for the bus is continuous and uniformly distributed between 4 minutes and 18minutes. What is the probability that it takes Susan less than 5 minutes given that it takes less than 11 minutes for her to wait for the busAssume that n independent count variables {X,,X2,...,X,} are identically distributed as X, ~ Poi(0) where i= 1,2,...,n and to estimate E(X,)= 0, consider the sample mean estimator ô = -5x,. n (b) Use the expectation of the distribution of S = nô to compute the expectation E(@) and the bias of this estimator. State whether this estimator unbiased. (c) Use the variance of the distribution of S= nô to examine var(Ô) and the standard error of this estimator. Use the variance and the bias of ê to compute the Mean Squared Error (MSE) of this estimator. Comment on the MSE behaviour in the asymptotic limit as n→o. State whether this estimator is consistent or (d) not.Compute P(X) using the binomial probability formulaThen determine whether the normal distribution can be ed to estimate this probabilityIf soapproximate P(X) using the normal distribution and compare the result th the exact probability n=53, p = 0.7 and X = 41
- c) Suppose that a random variable Y is uniformly distributed on an interval (0,1) and let c> 0 be a constant. i) Find the moment generating function of X = -cY. ii) What is the distribution of X? 2| PageLet Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1X is a normally distributed random variable with mean 72 and standard deviation 22. Use calculator to find the probability indicated. a) P(78<X<127) b) P(60<X<90) c) P(X=80) *continuous
