Manipulate[Column[{
   Plot[
    {PDF[BetaDistribution[alpha, beta], x], 
     CDF[BetaDistribution[alpha, beta], x]}, {x, 0, 1},
    PlotStyle -> {Blue, Automatic},
    PlotRange -> {-.03, 2}, 
    ImageSize -> 400,
    PlotLabel -> 
     Row[{Style["density function ", Blue], Style[" and ", Black], 
       Style["distribution function", Purple], 
       Style[" of a random variable~Beta(\[Alpha], \[Beta])", Black]}],
    Epilog -> {Thick, Red, 
      Line[{{Mean[BetaDistribution[alpha, beta]], 0}, 
         {Mean[BetaDistribution[alpha, beta]], 
         Max[PDF[BetaDistribution[alpha, beta], 
           Mean[BetaDistribution[alpha, beta]]], 
          CDF[BetaDistribution[alpha, beta], 
           Mean[BetaDistribution[alpha, beta]]]]}}]} ],
   Grid[{
     {Text@Style["mean: ", 12], 
      Text@Mean[BetaDistribution[\[Alpha], \[Beta]]], " = ", 
      Text@NumberForm[
        Mean[BetaDistribution[alpha, beta]], {6, 3}]}, {Text@
       Style[" variance: ", 12], 
      Text@Variance[BetaDistribution[\[Alpha], \[Beta]]], " = ", 
      Text@
       NumberForm[Variance[BetaDistribution[alpha, beta]], {6, 3}]}},
     Alignment -> {Right, Center}
    ]}, ItemSize -> {35, 5}, Alignment -> Center],
 {{alpha, 2., "parameter \[Alpha]"}, 0.1, 3, 
  Appearance -> "Labeled"},
 {{beta, 2., "parameter \[Beta]"}, 0.1, 3, Appearance -> "Labeled"}]