STA 312 Continuous Probability Distributions Notes

Problem 1

  • Continuous Random Variable (r.v.): The variable X represents the reaction time from a stimulus in seconds.

  • Probability Density Function (pdf):

  • The pdf for X is given as:
    [ f(x) = kx^2 \quad \text{ for } 1 \leq x \leq 6 \quad \text{ and } \quad f(x) = 0 \quad \text{ otherwise} ]

  • Goal: Find the value of k such that the function is a valid pdf.

  • Criteria for a Valid pdf:

  • The integral of the pdf over its range must equal 1:
    [ \int_{1}^{6} f(x) \, dx = 1 ]

  • Integrating the pdf:

  • [ \int{1}^{6} kx^2 \, dx = k \left[ \frac{x^3}{3} \right]{1}^{6} = k \left( \frac{6^3}{3} - \frac{1^3}{3} \right) = k \left( 72 - \frac{1}{3} \right) = k \left( \frac{215}{3} \right) ]

  • Set this equal to 1 and solve for k:
    [ k \cdot \frac{215}{3} = 1 \Rightarrow k = \frac{3}{215} ]

  • Graphing the pdf:

  • The x-axis ranges from 1 to 6, and the y-axis will range according to the value of k obtained.

  • Ensure proper scale and label both axes in the graph.

Problem 2

  • Continuous Random Variable (r.v.): The variable X signifies the number of inches of rain that falls.

  • Probability Density Function (pdf):

  • Given pdf is:
    [ f(x) = 2 - 2x \quad \text{ for } 0 < x < 1 \quad \text{ and } \quad f(x) = 0 \quad \text{ otherwise} ]

  • Finding Probabilities:
    a) Probability that over half an inch falls (P(X > 0.5)):
    [ P(X > 0.5) = \int_{0.5}^{1} (2 - 2x) \, dx ]

    b) Probability that less than 0.4 inches falls (P(X < 0.4)):
    [ P(X < 0.4) = \int_{0}^{0.4} (2 - 2x) \, dx ]

    c) Probability that less than 1.2 inches falls (P(X < 1.2)):

    • Since the pdf is defined only for 0 < x < 1, P(X < 1.2) = 1 (as it is the total area under the pdf from 0 to 1).

Problem 4

  • Continuous Random Variable (r.v.): The variable X has the following probability density function:

  • [ f(x) = 7 \sqrt{x - 1} \quad \text{ for } 1 \leq x \leq 4 \quad \text{ and } \quad f(x) = 0 \quad \text{ otherwise} ]

  • Finding the Cumulative Distribution Function (cdf):

  • The cdf, F(x), is obtained by integrating the pdf:
    [ F(x) = \int_{1}^{x} f(t) \, dt \quad \text{ for } 1 \leq x \leq 4 ]

  • Graphing the cdf:

  • Label axes clearly, x-axis for values of X, y-axis for F(x).

  • Scale appropriately to depict behavior of the cdf from x = 1 to x = 4.

  • Using the cdf for probabilities:
    1) Probability P(X > 1.8):
    [ P(X > 1.8) = 1 - F(1.8) ]
    2) Probability P(X = 3):

    • For continuous random variables, P(X = x) = 0 for any specific value x.
      3) Probability P(2.25 < X < 4):
      [ P(2.25 < X < 4) = F(4) - F(2.25) ]