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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .Suppose that the unknown X is ≥ 2 and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?Find (a) the mean of the distribution, (b) the standard deviation of the distribution, and (c) the probability that the random variable is between the mean and 1 standard deviation above the mean The length of time (in years) until a particular radioactive particle decays is a random variable t with probability density function defined by ƒ(t) = 4e-4t for t in [0, ∞].
- If X is a continuous variable in the range 3 > X > 0 and its distribution function is as follows: F ( x ) = k : ( x3 + x2) find the probability density function?Verify that p(x) = 3x - 4 is a probability density function on [1, oo)and calculate its mean value.2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)
- Find a value of k that will make f a probability density function on the indicated interval.ƒ(x) = kx3; [2, 4]Suppose that ƒ is a uniform joint probability density function on0 ≤ x 6 2, 0 ≤ y < 3. What is the formula for ƒ? What is theprobability that X < Y?Find a value of k that will make f a probability density function on the indicated interval. ƒ(x) = kx; [2, 4]
- If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).Find a value of k that will make f a probability density function on the indicated interval.ƒ(x) = kx; [2, 3]Find the conditional expectation E(Y/X=0.47) if the joint probability density function of the random variable X and Y isf(x, y) = 1/x, 0 < y ≤ x ≤ 1.