Let the probability function of the random variable X be defined as f(x)=cx(1-x) for 0

Q: . Determine the value c so that each of the following functions can serve as probability…

A: Find the value of constant value (c)

Q: Let the probability function of the random variable X be defined as f(x)=1/(2(1+∣x∣)²). Draw the…

A: From the given information, Consider, the pdf: f(x)=1/(2(1+∣x∣)²) fx={ 121+x2, x≥0      ={121-x2,…

Q: The probability distribution function of the discrete random variable X is x -1 0 2 5 P(X=x) 0.10…

A: Given :   X -1 0 2 5 P (x) 0.10 0.50 0.30 0.10

Q: Let X be a discrete random variable with the following probability distribution: X = 0 | P(X = 0) =…

A: The discrete value of x and their probability are, X 0 1 2 3 4 5 6 P(X=x) 0.1 0.15 0.2 0.25…

Q: The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise…

A: Given information: The probability distribution function of the random variable X is defined as:…

Q: (a) Find the expectation E (X) of X. E(X)= (b) Find the variance Var(X) of X. Var (x) =

A: Given: Pmf of X is X P(X=x) -20 0.10 -10 0.05 0 0.40 10 0.10 20 0.30 30 0.05 we…

Q: (a) Find the expectation E (X) of X. E (X) = [] (b) Find the variance Var(X) of X. Varlx) = D

A: The following table shows the provided outputs of the discrete random variables, along with the…

Q: Determine f(2)  using the indicated probability distribution of the discrete random variable X…

A:

Q: The continuous random variable X has a cumulative distribution function F(x) given by: x 2. Find…

A: The CDF of X is given by, Fx = 0x&lt;1kx4+x2-21≤x≤21x&gt;2 Now, X is continuous random variable. So,…

Q: Let X be the random variable with the cumulative probability distribution: 0 F(x)= 1 = {xx ²₁ x 2…

A:

Q: a) Find the expectation E (X) of X. E (x) = | %3D (b) Find the variance Var(X) of X. Var (x) = 0

A: Here given discrete probability distribution of x  Here use expected value and Variance formula

Q: Find the expected value E(X), E(X2) and variance (using the definition formula) if X is a discrete…

A: Given : X P (X) 0 14 2 18 -2 14 4 18 -4 14

Q: Understand the probability distribution of a random variable, based on a real-world situation, and…

A: Given :  X P(X) 200 0.6 500 0  Expected value ,E(X) = μ=∑X.P(X)…

Q: a) Find the expectation E (X) of X. E (x) = [] %3D b) Find the variance Var(X) of X. Var(X) Var (X)…

A: Consider the table ,  x P(x) x*P(x) x^2*P(x) 30 0.05 1.5 45 40 0.3 12 480 50 0.1 5 250 60…

Q: The probability distribution function of a discrete random variable X is x2 55 ,x = 1,2,3,4,5 0,…

A: Find  Var(3X+2)=? 5E(2X-1)=? E(2X+1)2=?

Q: (a) Find the expectation E (X) of X. E (x) = | %3D (b) Find the variance Var(X) of X. Var(X) = 0 %3D

A: Given: x p(x) 40 0.05 50 0.25 60 0.25 70 0.05 80 0.35 90 0.05 Total 1

Q: A discrete random variable X has the probability distribution as follows: *(x)= 5-x ;x%- 1,2,3,4 10…

A:

Q: Find the Mean of f(x), where f(x) = 6X+6. The probability distribution of a random variable X ; if X…

A: Mean is an average value around which the whole data set is concentrated.…

Q: (a): Find the probability distribution function fy(x) for the random variable X whose distribution…

A: We have to find probability distribution function

Q: Let X be a random variable and f a Borel function. Suppose that f (X) is not a.s. constant. Is it…

A: Here X is a random variable and f is a Borel function. Lets suppose that f(X) is not a constant. Let…

Q: Find the mean, variance and standard deviation of the discrete random variable X whose probability…

A:

Q: Let X be a random variable distributed exponentially with parameter such that is 1/2 find the…

A:

Q: . The cumulative distribution function of a discrete random variable X is x< 0 1 0 <x<1 3 7 F(x) = |…

A: From the given information, Fx=0       x&lt;013    0≤x&lt;1712    1≤x&lt;21724    2≤x&lt;31…

Q: Let X be a discrete random variable with expectation 1.5 and variance 2. Let Y = -2 X+9. Find the…

A:

Q: The probability distribution of random variable X is defined as follows:   X 0 1 2 3 4…

A: The expected value is calculated by the product of each observation and the corresponding…

Q: Let the function P {(1, 2)} = P {(1, 3)} = P {(2,2)} = P {(2,3)} = 1/6, P {(3, 3) } = 2/6.…

A: a) The associated bivariate distribution function   X=1 X=2 X=3 P(Y=y) Y=2  1/6  1/6 0  2/6…

Q: Let probability distribution function of X be defined by x = 0,1, 2, 3. P(x) = %3D otherwise.

A: Given, the probability distribution function of X be defined by…

Q: Let the probability function of the random variable X be defined as f(x)=cx^4 for x > 1, otherwise…

A: let the probabilty function is ,f(x)=Cx4      x&gt;1             0         otherwise

Q: x < 0 0 < x < 2 2 F (x) = 3x 8 2 1 2 < x

A: The cumulative distribution function of the continuous random variable X is given by F(x)  is a…

Q: The marginal distributions of two random variables X and Y, together with some values of their joint…

A: Solution-: We have following table:   Y=0 Y=1 Y=3 Total X=1   0.06   0.60 X=3   0.04   0.40…

Q: Let X be a continuous random variable with probability distribution as shown below: (x*(2x +), if0 <…

A:

Q: Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables having…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: (a) Find the expectation E (X) of X. E (x) = [] (b) Find the variance Var(X) of X. Var(x) = D %3D

A: Here given discrete probability distribution of X

Q: The marginal distributions of two random variables X and Y, together with some values of their joint…

A: Given  P(X=0)=0.5, P(X=1)=0.3, P(X=2)=0.2 P(Y=1)=0.5, P(Y=3)=0.5  Result: If X and Y are independent…

Q: Let the probability function of the random variable X be f(x) = {  x/45 if x = 1, 2, 3, ⋯ ⋯ ,9…

A:

Q: Let the life span of a material be given by the probability function f(x)=c²xe^(-cx)  (x≥ 0).…

A: Probability function  f(x) = c2xe-cx       x≥0

Q: A random variable X has the following pdf: f(x) = 9xe-3";x > (a) Calculate the expected value of the…

A: Let X be a continuous random variable taking values in interval -∞,∞ with probability density…

Q: Let the probability distribution function (pdf) for a continuous random variable be defined as f(x)…

A: Given, The probability distribution function for a continuous random variable be defined as,

Q: 1. Determine the constant c so that the function p(x) = 0(x) defines the probability den- sity for a…

A: From the given information, px=φxcx2+x6      =cx2+x6, x&gt;0 Condition for valid density function is…

Q: Determine the cumulative distribution function for the random variable with the probability density…

A:

Q: (a) Find the conditional mean and variance of X1, given X2 = x2, 0 < x2 < 1 (b) Find the…

A: Solution:

Q: The distribution function of a random variable X is given by 0, for x 1. Find (a) P(-< X < }); (b)…

A: From the given information, Distribution function can be expressed as: Now PDF can be drawn from…

Q: A- Let x be a discrete random variable with probability distribution function f(x)=k( x2 +20) and x=…

A:

Q: Let X1,.., Xn, be a random sample from the following probability distribution x+1 fX(x)=0De-*/", x >…

A: Solution:  a)  fX(x)  = x+1θ(θ+1) e-xθ             : x &gt;0 and θ&gt; 0 Joint pdf is given by,…

Q: let X be a random variable with the following probability distrubution -1 0.4 0 0.05 1 0.30 2…

A:

Q: Let the probability function of the random variable X be defined as f (x) = cx² for 0≤x≤1, otherwise…

A: Let X is the random variable: fXx=cx20≤x≤10otherwise a). The value of constant c: ∫-∞∞fXudu=1 So:…

Q: Let Y Bernoulli(X) where X is a random “probability of success" parameter. (a) Show that E(Y) = E(X)…

A:

Q: c) Let X be the random variable with the cumulative probability distribution: x < 0 F(x) = {₁ - e-²x…

A:

Q: Let X be a normal random variable, with E(X) =10, E(X^2)=200. Find the value c ,where c>0, such that…

A: Given Data; E(X)=10 V(X)=E(X^2)-E(X)^2=200-100=100 SD(X)=10   X~ Normal (10, 102)

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!